Discussion:
New trigonometric identity
Vinicius Claudino Ferraz
2020-11-02 12:42:33 UTC
https://pbs.twimg.com/media/El0Z7KvWoAUAqU5?format=png&name=900x900

\begin{align}
\tan^2 \biggl(\cfrac{\pi}{3} + \cfrac{1}{3} \text{ arctan } |x|\biggl) = 2 + 3x^2 + 2 \sqrt{x^2 + 1}\sqrt{9 x^2 + 1} \cos\biggl(\cfrac{1}{3} \text{ arctan } \cfrac{8|x|}{27 x^4 + 18 x^2 - 1}\biggl)
\end{align}

sketch of proof:

m = tan θ
v = tan 3θ = (3m - m³)/(1 - 3m²)
v/m =1/3 - 8/(9m² - 3)
x = m²
x³ + px + q = 0
Cardano
William Elliot
2020-11-03 09:42:16 UTC
Post by Vinicius Claudino Ferraz
https://pbs.twimg.com/media/El0Z7KvWoAUAqU5?format=png&name=900x900
\begin{align}
\tan^2 \biggl(\cfrac{\pi}{3} + \cfrac{1}{3} \text{ arctan } |x|\biggl) = 2 + 3x^2 + 2 \sqrt{x^2 + 1}\sqrt{9 x^2 + 1} \cos\biggl(\cfrac{1}{3} \text{ arctan } \cfrac{8|x|}{27 x^4 + 18 x^2 - 1}\biggl)
\end{align}
Unreadable. Use plain text to state the theorem.
Post by Vinicius Claudino Ferraz
m = tan Îž
v = tan 3Îž = (3m - mÂ³)/(1 - 3mÂ²)
v/m =1/3 - 8/(9mÂ² - 3)
x = mÂ²
xÂ³ + px + q = 0
Cardano
sobriquet
2020-11-03 11:25:54 UTC
Post by Vinicius Claudino Ferraz
https://pbs.twimg.com/media/El0Z7KvWoAUAqU5?format=png&name=900x900
\begin{align}
\tan^2 \biggl(\cfrac{\pi}{3} + \cfrac{1}{3} \text{ arctan } |x|\biggl) = 2 + 3x^2 + 2 \sqrt{x^2 + 1}\sqrt{9 x^2 + 1} \cos\biggl(\cfrac{1}{3} \text{ arctan } \cfrac{8|x|}{27 x^4 + 18 x^2 - 1}\biggl)
\end{align}
m = tan θ
v = tan 3θ = (3m - m³)/(1 - 3m²)
v/m =1/3 - 8/(9m² - 3)
x = m²
x³ + px + q = 0
Cardano
It doesn't seem to hold. If you substitute 0 for x, the left side is 3 and the right side is 4.

https://www.wolframalpha.com/input/?i=%28tan%28%28pi%2F3%29%2B%281%2F3%29arctan%28abs%280%29%29%29%29%5E2

https://www.wolframalpha.com/input/?i=2%2B3+0%5E2%2B2sqrt%280%5E2%2B1%29sqrt%289+0%5E2%2B1%29cos%28%281%2F3%29arctan%288%28abs%280%29%29%2F%2827+0%5E4%2B18+0%5E2-1%29%29%29

https://www.wolframalpha.com/input/?i=%28tan%28%28pi%2F3%29%2B%281%2F3%29arctan%28abs%28x%29%29%29%29%5E2%3D2%2B3x%5E2%2B2sqrt%28x%5E2%2B1%29sqrt%289x%5E2%2B1%29cos%28%281%2F3%29arctan%288%28abs%28x%29%29%2F%2827x%5E4%2B18x%5E2-1%29%29%29
Vinicius Claudino Ferraz
2020-11-03 13:32:32 UTC

sobriquet tried zero and it did not hold. well, zero is as dangerous as pi/2... I did not define the domain hahaha

Try x = 1.

william elliot said it is unreadable.

For 1 the formula says that

tan² 75° = 5 + 4 sqrt(5) cos(1/3 arctan 2/11)

The angle arctan 2/11 is constructable and its third part is constructable with ruler and compass.

For other ecses, I've just posted that

tan²( pi/3 + 1/3 arctan |x| ) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 ARCTAN 8|x|/(27x^4 + 18x² - 1) )

where ARCTAN is positive. if it were negative, add pi to it.

Now it holds for all x in the set of real numbers.

x = 0 ==> tan² 60° = 3 = 2 + 2 cos( 1/3 pi ) = 2 + 1

I think it will never hold for negatives and complex numbers because sqrt(x²) = |x| >= 0.

Sincerely,

Vinicius
the twenty twenty the November the third
Vinicius Claudino Ferraz
2020-11-03 13:45:13 UTC
https://www.wolframalpha.com/input/?i=5+%2B+4+*+sqrt%285%29+*+cos%281%2F3+*+arctan%282%2F11%29%29+-+tan+%2875*pi%2F180%29%5E2

tan²( pi/3 + 1/3 arctan |x| ) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 ARCTAN 8|x|/(27x^4 + 18x² - 1) )

where ARCTAN is positive. if it were not positive, add pi to it. So its image is (0, pi].
Vinicius Claudino Ferraz
2020-11-03 14:15:06 UTC
List of constructible lengths

a + b, ab, a/b, sqrt(a)

List of constructible angles

3 * pi/180 * k ∈ {0, 1, 2, ..., 119, 120}
arctan b/a, given two constructible lengths
a/2, bissectrix

Now add to this Field eight cubic roots or trisections:

x = tan 0°
1) x = tan 9°
2) x = tan 18°
3) x = tan 27°
4) x = tan 36°
5) x = tan 45° yields (11 + 2i)^(1/3) ~ construction of the angle 1/3 arctan 2/11
6) x = tan 54°
7) x = tan 63°
8) x = tan 72°
sobriquet
2020-11-03 16:45:43 UTC
Post by Vinicius Claudino Ferraz
sobriquet tried zero and it did not hold. well, zero is as dangerous as pi/2... I did not define the domain hahaha
Try x = 1.
william elliot said it is unreadable.
For 1 the formula says that
tan² 75° = 5 + 4 sqrt(5) cos(1/3 arctan 2/11)
The angle arctan 2/11 is constructable and its third part is constructable with ruler and compass.
For other ecses, I've just posted that
tan²( pi/3 + 1/3 arctan |x| ) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 ARCTAN 8|x|/(27x^4 + 18x² - 1) )
where ARCTAN is positive. if it were negative, add pi to it.
Now it holds for all x in the set of real numbers.
x = 0 ==> tan² 60° = 3 = 2 + 2 cos( 1/3 pi ) = 2 + 1
I think it will never hold for negatives and complex numbers because sqrt(x²) = |x| >= 0.
Sincerely,
Vinicius
the twenty twenty the November the third
https://www.desmos.com/calculator/8fcyum02zf

It fails to hold for small numbers close to zero.
Vinicius Claudino Ferraz
2020-11-03 19:21:17 UTC
Post by sobriquet
It fails to hold for small numbers close to zero.
That's because for small numbers you have to add pi to the arctan.

Put there this condition

p(x) = 27x^4 + 18x² - 1
f(x) = tan²( pi/3 + 1/3 arctan |x| )
g(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 arctan 8|x|/p )
h(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( pi/3 + 1/3 arctan 8|x|/p )
q(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 * pi/2 )

if p(x) > 0 then f(x) = g(x)
if p(x) < 0 then f(x) = h(x)
if p(x) = 0 then f(x) = q(x)

27x^4 + 18x² - 1 < 0
-0.227083 < x < 0.227083 ~ the angle is between 12° and 13°

https://www.wolframalpha.com/input/?i=27x%5E4+%2B+18x%5E2+-+1+%3C%3D+0

I'm worried
1) 8|x|/p = y ==> x = T(y)
2) there are 8 cubic roots which are constructible. now I want the tangent of 5t.
3) as the polynomial of sqrt(2)/2 is 2x² - 1 = 0 I want the polynomials of cos 3° k, the constructible angles,
compare the polynomial of t = arctan 2/11 with t/3.

First I thought as if it were as Q(2^(1/3)) = {a + b * 2^(1/3) + c * 4^(1/3)}
but no. Simply, the cubic root has disappeared and has become square root.
mitchr...@gmail.com
2020-11-03 23:57:59 UTC
What about a triangle with an infinitely small angle?
that looks like lines that are infinitely close to parallel.
sobriquet
2020-11-04 00:35:54 UTC
Post by Vinicius Claudino Ferraz
Post by sobriquet
It fails to hold for small numbers close to zero.
That's because for small numbers you have to add pi to the arctan.
Ok, but that seems a bit like saying abs(x) = x holds although for
x < 0 you have to multiply x by -1 to make it hold.
So it's not entirely clear what it means for some equation to hold
if there is a range of cases where you have to tweak it a bit to
make it hold.
Post by Vinicius Claudino Ferraz
Put there this condition
p(x) = 27x^4 + 18x² - 1
f(x) = tan²( pi/3 + 1/3 arctan |x| )
g(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 arctan 8|x|/p )
h(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( pi/3 + 1/3 arctan 8|x|/p )
q(x) = 2 + 3x² + 2 sqrt{x² + 1} sqrt{9x² + 1} cos( 1/3 * pi/2 )
if p(x) > 0 then f(x) = g(x)
if p(x) < 0 then f(x) = h(x)
if p(x) = 0 then f(x) = q(x)
27x^4 + 18x² - 1 < 0
-0.227083 < x < 0.227083 ~ the angle is between 12° and 13°
https://www.wolframalpha.com/input/?i=27x%5E4+%2B+18x%5E2+-+1+%3C%3D+0
I'm worried
1) 8|x|/p = y ==> x = T(y)
2) there are 8 cubic roots which are constructible. now I want the tangent of 5t.
3) as the polynomial of sqrt(2)/2 is 2x² - 1 = 0 I want the polynomials of cos 3° k, the constructible angles,
compare the polynomial of t = arctan 2/11 with t/3.
How do you associate numbers like sqrt(2)/2 with (roots of) polynomials
like 2x² - 1 = 0, given that we might as well associate them with other polynomials that share the same root (like 2^(3/2)x^3 - 1 = 0 )?
Post by Vinicius Claudino Ferraz
First I thought as if it were as Q(2^(1/3)) = {a + b * 2^(1/3) + c * 4^(1/3)}
but no. Simply, the cubic root has disappeared and has become square root.
Vinicius Claudino Ferraz
2020-11-04 21:33:27 UTC
Post by sobriquet
Ok, but that seems a bit like saying abs(x) = x holds although for
x < 0 you have to multiply x by -1 to make it hold.
So it's not entirely clear what it means for some equation to hold
if there is a range of cases where you have to tweak it a bit to
make it hold.
So, let's just say the same language, guy. It holds only for x >=0, if you want to drop the abs(x). of course!

The cases are very common in mathematics, don't you like them?

We can first define F : ℝ → ℝ
x ↦ g(x), if and only if p(x) > 0
x ↦ h(x), if and only if p(x) < 0
x ↦ q(x), if and only if p(x) = 0

And second state the theorem: f(x) = F(x), for all x in ℝ.
Post by sobriquet
How do you associate numbers like sqrt(2)/2 with (roots of) polynomials
like 2x² - 1 = 0, given that we might as well associate them with other polynomials that share the same root (like 2^(3/2)x^3 - 1 = 0 )?
That's the chapter "minimal polynomial". It's calculable.
https://www.wolframalpha.com/input/?i=MinimalPolynomial%28cos%2836*pi%2F180%29%2Cx%29

Definition: the minimal polynomial of x₀ is p(x) if and only if p(x₀) = 0, deg p(x) is the minimum possible, p(x) has integer coefficients, and the leader coefficient is positive.

I've just got the vision:
x₀ = 1 ⇒ f(x₀) = 2/11
x₁ = 2/11 ⇒ f(x₁) = another constructible thing = x₂
This yields an INFINITE sequence.
Such that t = 1/3 arctan x_n is constructible with ruler and compass.

Sincerely,

Vinicius
20.20.Nov.04th
Vinicius Claudino Ferraz
2020-11-05 13:49:50 UTC
Second ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%28sqrt%285+%2B+2*sqrt%285%29%29%2F%28163+%2B+72*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F31+%28219+%2B+41+sqrt%285%29+%2B+sqrt%286+%283065+%2B+1189+sqrt%285%29%29%29%29%29

Now, let's take the simmetry with the inverse of the parameter:

x = 1/y for instance y = 2/11 => x = 11/2
f(x) = 8x/(27x^4 + 18x² - 1)
f(1/y) = 8/y (27/y^4 + 18/y^2 - 1)
f(1/y) = 8(27 + 18y^2 - y^4)/y^5

It was 1 unique sequence (x_n), now we have the second sequence y_n = 1/x_n .
Vinicius Claudino Ferraz
2020-11-05 20:27:54 UTC
third ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29

fourth ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29

I've been thinking about x = z = r(cos t + i sen t)
and if 0 <= t < pi then abs(x) := z
else if pi <= t < 2 pi then abs(x) := -z (opposite of negative, for example)
because it came from sqrt(x^2). it "maybe" will hold in half plane.

Somebody please teach me. According to wikipedia,
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane

arctan(2i) = integral from 0 to 1 of dz/(1 + z^2) + integral from 1 to 2i of dz/(1 + z^2)

The first is arctan(1) = pi/4. And the second?

I forgot how to integrate f(z) dz from a to b.
Everything used to end up in Cauchy and number of holes.

Vinicius
mitchr...@gmail.com
2020-11-05 20:34:42 UTC
Post by Vinicius Claudino Ferraz
third ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29
fourth ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29
I've been thinking about x = z = r(cos t + i sen t)
and if 0 <= t < pi then abs(x) := z
else if pi <= t < 2 pi then abs(x) := -z (opposite of negative, for example)
because it came from sqrt(x^2). it "maybe" will hold in half plane.
Somebody please teach me. According to wikipedia,
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane
arctan(2i) = integral from 0 to 1 of dz/(1 + z^2) + integral from 1 to 2i of dz/(1 + z^2)
The first is arctan(1) = pi/4. And the second?
I forgot how to integrate f(z) dz from a to b.
Everything used to end up in Cauchy and number of holes.
Vinicius
Trig is your first polygon. That polygon is also QM...
Vinicius Claudino Ferraz
2020-11-05 20:43:48 UTC
Post by ***@gmail.com
Trig is your first polygon. That polygon is also QM...

I know there is a straight line from (1,0) to (p,q).
y = q(x - 1)/(p - 1)
I want line integral from 1 to p + qi.
mitchr...@gmail.com
2020-11-05 21:00:56 UTC
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
Trig is your first polygon. That polygon is also QM...
There is a first polygon. When it is right angle it is both QM and Trig...
Vinicius Claudino Ferraz
2020-11-05 21:14:24 UTC
Post by ***@gmail.com
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
Trig is your first polygon. That polygon is also QM...
There is a first polygon. When it is right angle it is both QM and Trig...
Among all polygons, who is the first?
Trig seems to qualify... Define that property, if it's meaningful.
QM for me is a segment. Where are the points Q and M? ; - P

At least we have countably many trisections. Luckily, it's not uncountable case.
Maybe we should "complete" that set by some way.
Ross A. Finlayson
2020-12-01 04:21:31 UTC
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
Trig is your first polygon. That polygon is also QM...
There is a first polygon. When it is right angle it is both QM and Trig...
Among all polygons, who is the first?
Trig seems to qualify... Define that property, if it's meaningful.
QM for me is a segment. Where are the points Q and M? ; - P
At least we have countably many trisections. Luckily, it's not uncountable case.
Maybe we should "complete" that set by some way.
(I don't know it but it reminds me tri-lateral-ometry that basically
after sine/cosine as the orthogonal and for the Pythagorean, that
it is the un-hinged regular n-gon, or n-lateral, that unfolding the
endpoint sweeps sine then folding back up and unfolding again
sweeps the other half of the period of sine, about what are all
defined as the functions in n-lateral-ometry past the trilateral's,
that each regular n-gon defines an n-lateral that happens to
trace out first sine/cosine then other functions like them, each
with all the results what follow like for Fourier make for summing
the terms of for orthogonal functions, that all real functions have
a way to so be written as a series in powers of those. Also then
it reminds me of the quadratic sieve but I don't know. Here for
computing the residue, or for pull-back or along those lines in the
complex, here it's nice that you are finding these terms as what
are basically modular there, it seems there suffices enough modular
terms in the prime-counting, that you are finding these expressions,
of the variables that result in terms that under usual algebraic
manipulations: appear to drop out as either "dimensionless", or,
not just like multiplying by 1/1, i.e. still being algebraic.)
Vinicius Claudino Ferraz
2020-12-01 22:11:35 UTC
Em terça-feira, 1 de dezembro de 2020 às 01:21:41 UTC-3, Ross A. Finlayson escreveu:
Fourier is by the way. An infinite sum of sines and cossines.
A kind of transform of functions.
But here we are dividing alpha /p and the result is a pretty: cos(p arccos(beta))

No link tonight. Everything is the same. Including the Spirits that make me owned LOL.
They are the same legion since 1982 when I reincarnated.

# Dividing quaternion by complex is not working. I need a better def powm(T, z)...

Below 4 + 6i + 10j + 37k ; p = 3 ; i^2 = j^2 = k^2 = -1

delta = [ (4.0 + 6.0j) (10.0 + 37.0j)]
[(-10.0 + 37.0j) (4.0 - 6.0j)]
m := delta*pi/180 = [(0.06981317008 + 0.1047197551j) (0.1745329252 + 0.6457718232j)]
[(-0.1745329252 + 0.6457718232j) (0.06981317008 - 0.1047197551j)]
a = [ (-0.8160820057 + 0.01219775137j) (0.02032958562 + 0.07521946678j)]
[(-0.02032958562 + 0.07521946678j) (-0.8160820057 - 0.01219775137j)]
b = [(0.2616975215 - 0.214297521j) (-0.3571625349 - 1.321501379j)]
[(0.3571625349 - 1.321501379j) (0.2616975215 + 0.214297521j)]
Re z = [(0.2572946101 - 0.153964933j) (-0.2566082216 - 0.94945042j)]
[ (0.2566082216 - 0.94945042j) (0.2572946101 + 0.153964933j)]
Im z = [(-0.9567507312 - 0.03246906898j) (-0.05411511496 - 0.2002259253j)]
[ (0.05411511496 - 0.2002259253j) (-0.9567507312 + 0.03246906898j)]
Mod z = [ (-0.186167613 + 0.04592389502j) (0.07653982503 + 0.2831973526j)]
[(-0.07653982503 + 0.2831973526j) (-0.186167613 - 0.04592389502j)]
cos Arg z = [(-2.796572984 + 0.1371635402j) (0.2286059003 + 0.845841831j)]
[(-0.2286059003 + 0.845841831j) (-2.796572984 - 0.1371635402j)] = [fraction] beta/gamma^p sometimes
sin Arg z = [(0.9426264608 + 0.4069351612j) (0.6782252686 + 2.509433494j)]
[(-0.6782252686 + 2.509433494j) (0.9426264608 - 0.4069351612j)]
tan Arg z = [(-0.03516496266 - 0.1472368196j) (-0.2453946994 - 0.9079603876j)]
[ (0.2453946994 - 0.9079603876j) (-0.03516496266 + 0.1472368196j)]
==> cos( [(3.0 + 0.0j) 0.0]
[ 0.0 (3.0 + 0.0j)] * arccos((tan^2(m)-a)/b)) = cos Arg z
delta = [ (124.0 + 6.0j) (10.0 + 37.0j)]
[(-10.0 + 37.0j) (124.0 - 6.0j)]
m := delta*pi/180 = [ (2.164208272 + 0.1047197551j) (0.1745329252 + 0.6457718232j)]
[(-0.1745329252 + 0.6457718232j) (2.164208272 - 0.1047197551j)]
a = [ (-0.8160820057 + 0.01219775137j) (0.02032958562 + 0.07521946678j)]
[(-0.02032958562 + 0.07521946678j) (-0.8160820057 - 0.01219775137j)]
b = [(0.2616975215 - 0.214297521j) (-0.3571625349 - 1.321501379j)]
[(0.3571625349 - 1.321501379j) (0.2616975215 + 0.214297521j)]
Re z = [(-0.2041383098 + 0.1007894302j) (0.1679823836 + 0.6215348195j)]
[(-0.1679823836 + 0.6215348195j) (-0.2041383098 - 0.1007894302j)]
Im z = [(-0.5890042647 - 0.02041656186j) (-0.03402760311 - 0.1259021314j)]
[ (0.03402760311 - 0.1259021314j) (-0.5890042647 + 0.02041656186j)]
Mod z = [ (-0.186167613 + 0.04592389502j) (0.07653982503 + 0.2831973526j)]
[(-0.07653982503 + 0.2831973526j) (-0.186167613 - 0.04592389502j)]
cos Arg z = [(1.884829414 - 0.07644037443j) (-0.1274006241 - 0.4713823091j)]
[(0.1274006241 - 0.4713823091j) (1.884829414 + 0.07644037443j)] = [fraction] beta/gamma^p sometimes
sin Arg z = [(0.5736246683 + 0.2511695785j) (0.4186159643 + 1.548879068j)]
[(-0.4186159643 + 1.548879068j) (0.5736246683 - 0.2511695785j)]
tan Arg z = [(0.07336041516 + 0.1362336953j) (0.2270561589 + 0.8401077879j)]
[(-0.2270561589 + 0.8401077879j) (0.07336041516 - 0.1362336953j)]
==> cos( [(3.0 + 0.0j) 0.0]
[ 0.0 (3.0 + 0.0j)] * arccos((tan^2(m)-a)/b)) = cos Arg z

# y = 2bd
# z = 2cd
# t = a**2 + b**2 + c**2 - d**2
# w = a**2 + b**2 + c**2 + d**2
#then: x^2 + y^2 + z^2 + t^2 == w^2
Vinicius Claudino Ferraz
2020-12-01 22:16:03 UTC
Why do the children cry when we say that zero is equal to 2*pi rad?
Why do other children cry when we say that 20 degrees are undiscovered.
Why do the kids cry when we say that 2^(1/12) must exist, but nobody discovered yet.
Maybe an extraterrestrial... Some kind of life that lasts time enough to discover.
We are not alone at the universe. Maybe an ET lives more than Matusalem and his 10240 biblical years having sex.
Vinicius Claudino Ferraz
2020-12-02 22:57:02 UTC
with 2 complexes, we make a quaternion, it could be 4x4
with 2 quaternions, we make an octonion 8x8
with 2 octonions, we make a sedenion 16x16
why is nobody interessed in 2 sedenions?
with them we make unnamed matrixes 32x32.
https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
Vinicius Claudino Ferraz
2020-12-04 21:56:34 UTC
https://www.wolframalpha.com/input/?i=24.48378444127528097477087281354090828809026731570800802127910383060111

the input is x = 24.48378444127528097477087281354090828809026731570800802127910383060111
the fucking output is x = 13584 - 6064 sqrt(5)
I need to implement that in Python.
Fast.
How does wolfram alpha do that disgrace?
Vinicius Claudino Ferraz
2020-12-05 20:04:03 UTC
https://www.wolframalpha.com/input/?i=cos%283*arccos%28%28tan%5E2%2812%2F1*pi%2F180%29-%2817-6*sqrt%285%29%29%29%2Fsqrt%281824-800*sqrt%285%29%29%29%29-%2813584-6064*sqrt%285%29%29%2F%28456-200*sqrt%285%29%29%5E1.5

Everything is intuitive here. 36 degrees / 3 = 12 degrees ~ sqrt(5) ~ golden number.
Vinicius Claudino Ferraz
2020-12-05 20:53:54 UTC
https://www.wolframalpha.com/input/?i=tan%5E2%285*pi%2F180%29%3D23-12*sqrt%283%29%2B2*sqrt%28944%2B-544*sqrt%283%29%29*cos%282*pi%2F3%2B1%2F3*arccos%28%2840960%2B-23648*sqrt%283%29%29%2F%28944%2B-544*sqrt%283%29%29%5E1.5%29%29

Everything should be intuitive here too. 15 degrees / 3 = 5 degrees. This angle must exist.

https://www.wolframalpha.com/input/?i=cos%283*arccos%28%28tan%5E2%285%2F1*pi%2F180%29-%2823%2B-12*sqrt%283%29%29%29%2F%282*sqrt%28944%2B-544*sqrt%283%29%29%29%29%29-%2840960%2B-23648*sqrt%283%29%29%2F%28944%2B-544*sqrt%283%29%29%5E1.5
Vinicius Claudino Ferraz
2020-12-07 22:57:35 UTC
48 lines is a little better than zero.
But it seems they are modulo 2. LOL

cos(3*arccos((tan^2(12/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(132/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(252/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(24/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(144/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(264/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(36/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(156/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(276/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(48/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(168/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(288/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(72/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(192/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(312/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(84/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(204/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(324/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(96/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(216/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(336/1*pi/180)-(17+6*sqrt(5)))/(2*sqrt(456+200*sqrt(5)))))-(13584+6064*sqrt(5))/(456+200*sqrt(5))^1.5
cos(3*arccos((tan^2(108/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(228/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(348/1*pi/180)-(17-6*sqrt(5)))/(2*sqrt(456-200*sqrt(5)))))-(13584-6064*sqrt(5))/(456-200*sqrt(5))^1.5
cos(3*arccos((tan^2(5/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(125/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(245/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(25/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(145/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(265/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(35/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(155/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(275/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(55/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(175/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(295/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(65/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(185/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(305/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(85/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(205/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(325/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(95/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(215/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(335/1*pi/180)-(23+12*sqrt(3)))/(2*sqrt(944+544*sqrt(3)))))-(40960+23648*sqrt(3))/(944+544*sqrt(3))^1.5
cos(3*arccos((tan^2(115/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(235/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5
cos(3*arccos((tan^2(355/1*pi/180)-(23-12*sqrt(3)))/(2*sqrt(944-544*sqrt(3)))))-(40960-23648*sqrt(3))/(944-544*sqrt(3))^1.5

Below the quaternion: 4 + 6i + 10j + 37k ; p = 3 ; i^2 = j^2 = k^2 = -1

delta = [ 4.0 -6.0 -10.0 -37.0]
[ 6.0 4.0 -37.0 10.0]
[10.0 37.0 4.0 -6.0]
[37.0 -10.0 6.0 4.0]
m := delta*pi/180 = [0.06981317008 -0.1047197551 -0.1745329252 -0.6457718232]
[ 0.1047197551 0.06981317008 -0.6457718232 0.1745329252]
[ 0.1745329252 0.6457718232 0.06981317008 -0.1047197551]
[ 0.6457718232 -0.1745329252 0.1047197551 0.06981317008]
a = [-0.8160820057 -0.01219775137 -0.02032958562 -0.07521946678]
[0.01219775137 -0.8160820057 -0.07521946678 0.02032958562]
[0.02032958562 0.07521946678 -0.8160820057 -0.01219775137]
[0.07521946678 -0.02032958562 0.01219775137 -0.8160820057]
b = [ 0.2616975215 0.214297521 0.3571625349 1.321501379]
[ -0.214297521 0.2616975215 1.321501379 -0.3571625349]
[-0.3571625349 -1.321501379 0.2616975215 0.214297521]
[ -1.321501379 0.3571625349 -0.214297521 0.2616975215]
Re z = [ 0.25729461 0.153964933 0.2566082216 0.9494504199]
[ -0.153964933 0.25729461 0.9494504199 -0.2566082216]
[-0.2566082216 -0.9494504199 0.25729461 0.153964933]
[-0.9494504199 0.2566082216 -0.153964933 0.25729461]
Im z = [ -0.9567507311 0.03246906896 0.05411511493 0.2002259252]
[-0.03246906896 -0.9567507311 0.2002259252 -0.05411511493]
[-0.05411511493 -0.2002259252 -0.9567507311 0.03246906896]
[ -0.2002259252 0.05411511493 -0.03246906896 -0.9567507311]
Mod z = [ -0.186167613 -0.04592389501 -0.07653982502 -0.2831973525]
[0.04592389501 -0.186167613 -0.2831973525 0.07653982502]
[0.07653982502 0.2831973525 -0.186167613 -0.04592389501]
[ 0.2831973525 -0.07653982502 0.04592389501 -0.186167613]
cos Arg z = [-2.796572984 -0.1371635402 -0.2286059004 -0.8458418315]
[0.1371635402 -2.796572984 -0.8458418315 0.2286059004]
[0.2286059004 0.8458418315 -2.796572984 -0.1371635402]
[0.8458418315 -0.2286059004 0.1371635402 -2.796572984] = [fraction] beta/gamma^p sometimes
sin Arg z = [0.9426264614 -0.4069351612 -0.6782252686 -2.509433494]
[0.4069351612 0.9426264614 -2.509433494 0.6782252686]
[0.6782252686 2.509433494 0.9426264614 -0.4069351612]
[ 2.509433494 -0.6782252686 0.4069351612 0.9426264614]
tan Arg z = [-0.03516496267 0.1472368196 0.2453946994 0.9079603876]
[ -0.1472368196 -0.03516496267 0.9079603876 -0.2453946994]
[ -0.2453946994 -0.9079603876 -0.03516496267 0.1472368196]
[ -0.9079603876 0.2453946994 -0.1472368196 -0.03516496267]
==> cos( [3.0 0.0 0.0 0.0]
[0.0 3.0 0.0 0.0]
[0.0 0.0 3.0 0.0]
[0.0 0.0 0.0 3.0] * arccos((tan^2(m)-a)/b)) = cos Arg z
delta = [124.0 -6.0 -10.0 -37.0]
[ 6.0 124.0 -37.0 10.0]
[ 10.0 37.0 124.0 -6.0]
[ 37.0 -10.0 6.0 124.0]
m := delta*pi/180 = [ 2.164208272 -0.1047197551 -0.1745329252 -0.6457718232]
[0.1047197551 2.164208272 -0.6457718232 0.1745329252]
[0.1745329252 0.6457718232 2.164208272 -0.1047197551]
[0.6457718232 -0.1745329252 0.1047197551 2.164208272]
a = [-0.8160820057 -0.01219775137 -0.02032958562 -0.07521946678]
[0.01219775137 -0.8160820057 -0.07521946678 0.02032958562]
[0.02032958562 0.07521946678 -0.8160820057 -0.01219775137]
[0.07521946678 -0.02032958562 0.01219775137 -0.8160820057]
b = [ 0.2616975215 0.214297521 0.3571625349 1.321501379]
[ -0.214297521 0.2616975215 1.321501379 -0.3571625349]
[-0.3571625349 -1.321501379 0.2616975215 0.214297521]
[ -1.321501379 0.3571625349 -0.214297521 0.2616975215]
Re z = [-0.2041383099 -0.1007894302 -0.1679823836 -0.6215348194]
[ 0.1007894302 -0.2041383099 -0.6215348194 0.1679823836]
[ 0.1679823836 0.6215348194 -0.2041383099 -0.1007894302]
[ 0.6215348194 -0.1679823836 0.1007894302 -0.2041383099]
Im z = [ -0.5890042646 0.02041656189 0.03402760317 0.1259021316]
[-0.02041656189 -0.5890042646 0.1259021316 -0.03402760317]
[-0.03402760317 -0.1259021316 -0.5890042646 0.02041656189]
[ -0.1259021316 0.03402760317 -0.02041656189 -0.5890042646]
Mod z = [ -0.186167613 -0.04592389501 -0.07653982502 -0.2831973525]
[0.04592389501 -0.186167613 -0.2831973525 0.07653982502]
[0.07653982502 0.2831973525 -0.186167613 -0.04592389501]
[ 0.2831973525 -0.07653982502 0.04592389501 -0.186167613]
cos Arg z = [ 1.884829414 0.07644037436 0.127400624 0.4713823087]
[-0.07644037436 1.884829414 0.4713823087 -0.127400624]
[ -0.127400624 -0.4713823087 1.884829414 0.07644037436]
[ -0.4713823087 0.127400624 -0.07644037436 1.884829414] = [fraction] beta/gamma^p sometimes
sin Arg z = [0.5736246678 -0.2511695786 -0.4186159644 -1.548879068]
[0.2511695786 0.5736246678 -1.548879068 0.4186159644]
[0.4186159644 1.548879068 0.5736246678 -0.2511695786]
[ 1.548879068 -0.4186159644 0.2511695786 0.5736246678]
tan Arg z = [0.07336041509 -0.1362336953 -0.2270561589 -0.8401077879]
[ 0.1362336953 0.07336041509 -0.8401077879 0.2270561589]
[ 0.2270561589 0.8401077879 0.07336041509 -0.1362336953]
[ 0.8401077879 -0.2270561589 0.1362336953 0.07336041509]
==> cos( [3.0 0.0 0.0 0.0]
[0.0 3.0 0.0 0.0]
[0.0 0.0 3.0 0.0]
[0.0 0.0 0.0 3.0] * arccos((tan^2(m)-a)/b)) = cos Arg z

Below the 'bi-quaternion': (3,4) + (6,8)i + (10,20)j + (37,39)k ; p = 3 ; i^2 = j^2 = k^2 = -1

delta = [ (3.0 + 4.0j) (-6.0 - 8.0j) (-10.0 - 20.0j) (-37.0 - 39.0j)]
[ (6.0 + 8.0j) (3.0 + 4.0j) (-37.0 - 39.0j) (10.0 + 20.0j)]
[(10.0 + 20.0j) (37.0 + 39.0j) (3.0 + 4.0j) (-6.0 - 8.0j)]
[(37.0 + 39.0j) (-10.0 - 20.0j) (6.0 + 8.0j) (3.0 + 4.0j)]
m := delta*pi/180 = [(0.05235987756 + 0.06981317008j) (-0.1047197551 - 0.1396263402j) (-0.1745329252 - 0.3490658504j) (-0.6457718232 - 0.6806784083j)]
[ (0.1047197551 + 0.1396263402j) (0.05235987756 + 0.06981317008j) (-0.6457718232 - 0.6806784083j) (0.1745329252 + 0.3490658504j)]
[ (0.1745329252 + 0.3490658504j) (0.6457718232 + 0.6806784083j) (0.05235987756 + 0.06981317008j) (-0.1047197551 - 0.1396263402j)]
[ (0.6457718232 + 0.6806784083j) (-0.1745329252 - 0.3490658504j) (0.1047197551 + 0.1396263402j) (0.05235987756 + 0.06981317008j)]
a = [ (-0.9823679967 + 0.2235804878j) (0.0162080435 - 0.01109890549j) (0.04009725789 - 0.01793437468j) (0.07966963086 - 0.06931714229j)]
[ (-0.0162080435 + 0.01109890549j) (-0.9823679967 + 0.2235804878j) (0.07966963086 - 0.06931714229j) (-0.04009725789 + 0.01793437468j)]
[(-0.04009725789 + 0.01793437468j) (-0.07966963086 + 0.06931714229j) (-0.9823679967 + 0.2235804878j) (0.0162080435 - 0.01109890549j)]
[(-0.07966963086 + 0.06931714229j) (0.04009725789 - 0.01793437468j) (-0.0162080435 + 0.01109890549j) (-0.9823679967 + 0.2235804878j)]
b = [ (1.0195121 - 1.129522723j) (-0.06875963585 - 0.004667902568j) (-0.1494040378 - 0.03777323999j) (-0.370070555 + 0.01770437451j)]
[(0.06875963585 + 0.004667902568j) (1.0195121 - 1.129522723j) (-0.370070555 + 0.01770437451j) (0.1494040378 + 0.03777323999j)]
[ (0.1494040378 + 0.03777323999j) (0.370070555 - 0.01770437451j) (1.0195121 - 1.129522723j) (-0.06875963585 - 0.004667902568j)]
[ (0.370070555 - 0.01770437451j) (-0.1494040378 - 0.03777323999j) (0.06875963585 + 0.004667902568j) (1.0195121 - 1.129522723j)]
Re z = [ (-0.3698273216 + 0.3013064752j) (-0.008724672152 - 0.05438227852j) (0.002558632624 - 0.1231308816j) (-0.08030676209 - 0.2849920705j)]
[(0.008724672152 + 0.05438227852j) (-0.3698273216 + 0.3013064752j) (-0.08030676209 - 0.2849920705j) (-0.002558632624 + 0.1231308816j)]
[(-0.002558632624 + 0.1231308816j) (0.08030676209 + 0.2849920705j) (-0.3698273216 + 0.3013064752j) (-0.008724672152 - 0.05438227852j)]
[ (0.08030676209 + 0.2849920705j) (0.002558632624 - 0.1231308816j) (0.008724672152 + 0.05438227852j) (-0.3698273216 + 0.3013064752j)]
Im z = [ (-0.5877099231 - 0.4384877699j) (0.04566808056 + 0.03927263299j) (0.08476072401 + 0.1046670248j) (0.2682165826 + 0.1814016501j)]
[(-0.04566808056 - 0.03927263299j) (-0.5877099231 - 0.4384877699j) (0.2682165826 + 0.1814016501j) (-0.08476072401 - 0.1046670248j)]
[ (-0.08476072401 - 0.1046670248j) (-0.2682165826 - 0.1814016501j) (-0.5877099231 - 0.4384877699j) (0.04566808056 + 0.03927263299j)]
[ (-0.2682165826 - 0.1814016501j) (0.08476072401 + 0.1046670248j) (-0.04566808056 - 0.03927263299j) (-0.5877099231 - 0.4384877699j)]
Mod z = [ (-0.416836924 - 0.190958951j) (0.003459564238 + 0.05988414779j) (-0.01634261779 + 0.1331289508j) (0.05560224466 + 0.3176364194j)]
[(-0.003459564238 - 0.05988414779j) (-0.416836924 - 0.190958951j) (0.05560224466 + 0.3176364194j) (0.01634261779 - 0.1331289508j)]
[ (0.01634261779 - 0.1331289508j) (-0.05560224466 - 0.3176364194j) (-0.416836924 - 0.190958951j) (0.003459564238 + 0.05988414779j)]
[ (-0.05560224466 - 0.3176364194j) (-0.01634261779 + 0.1331289508j) (-0.003459564238 - 0.05988414779j) (-0.416836924 - 0.190958951j)]
cos Arg z = [ (-0.3324620903 - 1.697245848j) (0.2425360862 - 0.04249407155j) (0.5505770182 + 0.003527476998j) (1.268796376 - 0.3772907155j)]
[ (-0.2425360862 + 0.04249407155j) (-0.3324620903 - 1.697245848j) (1.268796376 - 0.3772907155j) (-0.5505770182 - 0.003527476998j)]
[(-0.5505770182 - 0.003527476998j) (-1.268796376 + 0.3772907155j) (-0.3324620903 - 1.697245848j) (0.2425360862 - 0.04249407155j)]
[ (-1.268796376 + 0.3772907155j) (0.5505770182 + 0.003527476998j) (-0.2425360862 + 0.04249407155j) (-0.3324620903 - 1.697245848j)] = [fraction] beta/gamma^p sometimes
sin Arg z = [ (2.1443828 - 0.3224297845j) (0.04240410447 + 0.1917512319j) (0.01658853711 + 0.4388143521j) (0.3453236817 + 0.9976610339j)]
[(-0.04240410447 - 0.1917512319j) (2.1443828 - 0.3224297845j) (0.3453236817 + 0.9976610339j) (-0.01658853711 - 0.4388143521j)]
[(-0.01658853711 - 0.4388143521j) (-0.3453236817 - 0.9976610339j) (2.1443828 - 0.3224297845j) (0.04240410447 + 0.1917512319j)]
[ (-0.3453236817 - 0.9976610339j) (0.01658853711 + 0.4388143521j) (-0.04240410447 - 0.1917512319j) (2.1443828 - 0.3224297845j)]
tan Arg z = [ (0.07631256997 + 2.251756695j) (0.2072214583 - 0.001707164894j) (0.4565700741 + 0.0791328205j) (1.105504145 - 0.1375935647j)]
[(-0.2072214583 + 0.001707164894j) (0.07631256997 + 2.251756695j) (1.105504145 - 0.1375935647j) (-0.4565700741 - 0.0791328205j)]
[ (-0.4565700741 - 0.0791328205j) (-1.105504145 + 0.1375935647j) (0.07631256997 + 2.251756695j) (0.2072214583 - 0.001707164894j)]
[ (-1.105504145 + 0.1375935647j) (0.4565700741 + 0.0791328205j) (-0.2072214583 + 0.001707164894j) (0.07631256997 + 2.251756695j)]
==> cos( [3.0 0.0 0.0 0.0]
[0.0 3.0 0.0 0.0]
[0.0 0.0 3.0 0.0]
[0.0 0.0 0.0 3.0] * arccos((tan^2(m)-a)/b)) = cos Arg z

Delta(2):

v &= \cfrac{2m}{1 - m^2} \\
- vm^2 - 2m + v &= 0 \\
m^2 &= x \Rightarrow m = \sqrt{x} \\
v - vx &= 2 sqrt{x} \\
v^2(x^2 - 2x + 1) = 4x
v^2 x^2 -2 x(v^2 + 2) + v^2 = 0
Delta = 4 (v^2 + 2)^2 - 4 v^4 = 16(v^2 + 1)
x = [v^2 + 2 \pm 2 sqrt{v^2 + 1}]/v^2

a = 1 + 2v^2
s = pa
c = 1
r = Mod z = d/2 = 2v^2(1 + v^2)
sqrt(r) = v sqrt{2} sqrt{1 + v^2}

x &= s/2 + 2 \sqrt{r} \cos (t/2)
[v^2 + 2 \pm 2 sqrt{v^2 + 1}]/v^2 = 1 + 2v^2 + 2 sqrt{r} \cos (t/2)
[-v^4 + 1 - sqrt{v^2 + 1}]^2/v^4 = 2 v^2 (1 + v^2) \cos^2 (t/2)
cos Arg z = [-v^4 + 1 - E]^2/[v^6 * E^2] - 1

Delta(4):

w = f(v) como acima
x = f(sqrt(w))

E = sqrt{v^2 + 1}
w = (v^2 + 2 - 2 E)/v^2
x = [w + 2 - 2 sqrt{w + 1}]/w
x = 2v * [v - sqrt{2v^2 + 2 - 2 E}]/(v^2 + 2 - 2 E) + 1

a = 3 + 4v^2
c = 38 + 32 v^2
d = 6a^2 - c = 6(9 + 16v^4 + 24v^2) - 38 - 32v^2 = 96v^4 + 112 v^2 + 16
r = Mod z = (b/2)^4 = (d/p)^2 = d^2/16 = 2^4(6v^4 + 7v^2 + 1)^2

x = a + 2 sqrt[4](r) cos (t/4)
cos(4t) = 2cos^2(2t) - 1 = 2(2c^2 - 1)^2 - 1 = 2(4c^4 - 4c^2 + 1) - 1
2v * [v - sqrt{2v^2 + 2 - 2 E}]/(v^2 + 2 - 2 E) + 1 = 3 + 4v^2 + 2*2 sqrt{6v^4 + 7v^2 + 1} cos(t/4)
[v * [v - sqrt{2v^2 + 2 - 2 E}]/(v^2 + 2 - 2 E) - 1 - 2v^2]/[2 sqrt{6v^4 + 7v^2 + 1}] = c
cos Arg z = 8c^4 - 8c^2 + 1

Exercise. We have 2, 3 and 4. induction(p). "The world wants" cos Arg z as a function of p.
Vinicius Claudino Ferraz
2020-12-09 21:59:50 UTC
no.
if a_1 ~ a_2 (mod p) ~ a_3 (mod q) are brothers,
he who constructs the angle a_1, he also constructs the angles (a_n).
and vice versa
he who does not construct the angle a_1, he does neither construct the angles (a_n).

it's like Q and R - Q. The constructible is densely greater then the unconstrutible.

Vinicius Claudino Ferraz
2020/12/09th
Vinicius Claudino Ferraz
2020-12-27 16:22:27 UTC

playlist

www.claudino.webs.com /
2020-12-31 18:48:45 UTC
x^5 + rx + s = 0
This code in Python works: https://drive.google.com/file/d/1XxcsllczeqMo0EgA46PoAXwis4--qJuM/view?usp=sharing

I'm going to solve tan 5x as a function of tan x.

a = \tan 2t = \cfrac{2m}{1 - m^2} \\
b = \tan 3t = \cfrac{3m - m^3}{1 - 3m^2} \\
v = \tan 5t = \cfrac{a + b}{1 - ab} = \cfrac{2m(1 - 3m^2) + (3m - m^3)(1 - m^2)}{(1 - m^2)(1 - 3m^2) - 2m(3m - m^3)} \\
= \cfrac{m^5 - 10m^3 + 5m}{5m^4 - 10m^2 + 1} \\
m^5 - 5vm^4 - 10m^3 + 10vm^2 + 5m - v = 0
m^2 = x \Rightarrow m = \sqrt{x} \\
x^2\sqrt{x} - 5vx^2 - 10x\sqrt{x} + 10vx + 5\sqrt{x} - v = 0
x(x^2 - 10x + 5)^2 = v^2(5x^2 - 10x + 1)^2
x^5 + (- 20 - 25v^2)x^4 + (110 + 100v^2) x^3 + (- 100 - 110v^2) x^2 + (25 + 20v^2) x - v^2 = 0
EXERCISE: Tschirnhaus HERE.

I know the five roots of this quintic.
v = \tan \theta
x = \tan^2 ((|\theta| + k \pi)/5)
Vinicius Claudino Ferraz
2021-01-12 21:20:24 UTC
Utility:
z = a + bi
|z| = 1
tau = b/a
cos(1/p arctan tau) = Re w
w = z^(1/p)

We almost have how to evaluate cos(1/3 arctan tau) = f(tau)
We almost have how to evaluate cos(1/p arctan tau) = f(tau, p)

1/p = n
z = cos t + i sin t
tau = tan t
w = z^n
Re w = cos nt = f(cos t)
Chebyshev, cos 2t, cos 3t, cos 4t, ...
https://en.wikipedia.org/wiki/Chebyshev_polynomials#Examples

Exercise: amplify Chebyshev to real p // complex p.
cos(1/p * t) = f(cos t)
Vinicius Claudino Ferraz
2021-01-19 21:28:46 UTC
Registering: tan nt = p(tan t)
https://pbs.twimg.com/media/EsH_cxGW4AAz11w?format=jpg&name=medium

\begin{align*}
2 \le n \in \mathbb{N}, \forall \theta \in \text{GL}(4, \mathbb{C}) &\Rightarrow \sum_{i = 0}^n (-1)^i \binom{n}{i} (\tan \theta)^{n - i} (\tan n\theta)^{[1 - (-1)^i]/2} = 0
\end{align*}

out of charity, there is no salvation at all.

sobriquet
2020-11-06 15:26:40 UTC
Post by Vinicius Claudino Ferraz
third ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29
fourth ugly number
https://www.wolframalpha.com/input/?i=cos%281%2F3+*+arctan%285*sqrt%285+%2B+2*sqrt%285%29%29%2F%2890+%2B+41*sqrt%285%29%29%29%29+-+1%2F4+sqrt%281%2F11+%28119+-+5+sqrt%285%29+%2B+sqrt%2830+%2885+%2B+31+sqrt%285%29%29%29%29%29
Links for 3rd and 4th one are identical?
Post by Vinicius Claudino Ferraz
I've been thinking about x = z = r(cos t + i sen t)
and if 0 <= t < pi then abs(x) := z
else if pi <= t < 2 pi then abs(x) := -z (opposite of negative, for example)
because it came from sqrt(x^2). it "maybe" will hold in half plane.
Somebody please teach me. According to wikipedia,
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Extension_to_complex_plane
arctan(2i) = integral from 0 to 1 of dz/(1 + z^2) + integral from 1 to 2i of dz/(1 + z^2)
The first is arctan(1) = pi/4. And the second?
I forgot how to integrate f(z) dz from a to b.
Everything used to end up in Cauchy and number of holes.
Vinicius
Vinicius Claudino Ferraz
2020-11-08 14:25:41 UTC
Post by sobriquet
Links for 3rd and 4th one are identical?

Let me summarize my "paper", maybe somebody can help.

tan t = m
tan 2t = 2m/(1 - m^2)
tan 3t = v = \cfrac{3m - m^3}{1 - 3m^2} \\

We want m extracted as a function of m.
m^3 - 3vm^2 - 3m + v = 0 \\

Align 3 points as below, for trisecting some angles.

A = (a, 0) \\
B = (b, bm) \Rightarrow AB: y = bm\cdot\cfrac{x - a}{b - a} \\
C = \biggl( c, c \cdot \cfrac{3m - m^3}{1 - 3m^2} \biggl) \in AB \\
a = bc\cdot\cfrac{v - m}{cv - bm}\\
\cfrac{v}{m} = \cfrac{b}{c} \cdot \cfrac{c - a}{b - a)} = \cfrac{1}{3} - \cfrac{8}{9m^2 - 3} \\
m^2 = x \Rightarrow m = \sqrt{x} \\

x\sqrt{x} - 3vx - 3\sqrt{x} + v = 0 \\
v^2(3x - 1)^2 = x(x - 3)^2 \\
v^2(9x^2 - 6x + 1) = x^3 - 6x^2 + 9x \\
s = 6 + 9v^2 \\
q = 9 + 6v^2 \\
p = v^2 \\
x = y + s/3 \\
y^3 + Py + Q = 0 \\
P = q - s^2/3 = -3(9v^4 + 10v^2 + 1) \\
Q = -2/27 s^3 + q s/3 - p = -2(27v^6 + 45v^4 + 17v^2 - 1) \\
Delta = Q^2/4 + P^3/27 = - 64v^2(v^2 + 1)^2 \\
w = a + bi = r e^{it} \\
a = -Q/2 = (v^2 + 1) (27 v^4 + 18 v^2 - 1) \\
b = \sqrt{-\Delta} = 8|v|(v^2 + 1) < 0 \\
x = s/3 + 2 \sqrt[3]{r} \cos (t/3)

RENAME x TO tan²( pi/3 + 1/3 arctan |x| )
RENAME s/3 TO 2 + 3x²
RENAME \sqrt[3]{r} TO sqrt{x² + 1} sqrt{9x² + 1}
RENAME t TO arctan 8|x|/(27x^4 + 18x² - 1), BUT THE IMAGE IS IN (0, pi]

Replace x by tan 9k * pi/180.

HABEMUS UGLY NUMBERS
https://pbs.twimg.com/media/EmPUMiIXIAIeK3Z?format=jpg&name=medium

Let my tree be as follows:
a = 1 = tan 45°
aa = 2/11
ab = 11/2
aaa = 21296/5497
aab = 5497/21296
aba = 704/404003
abb = 404003/704
They are countable. Are they dense? Are they completeable?

Now we might go to fifth degree.

a = \tan 2t = \cfrac{2m}{1 - m^2} \\
b = \tan 3t = \cfrac{3m - m^3}{1 - 3m^2} \\
v = \tan 5t = \cfrac{a + b}{1 - ab} = \cfrac{2m(1 - 3m^2) + (3m - m^3)(1 - m^2)}{(1 - m^2)(1 - 3m^2) - 2m(3m - m^3)} \\
= \cfrac{m^5 - 10m^3 + 5m}{5m^4 - 10m^2 + 1} \\
m^5 - 5vm^4 - 10m^3 + 10vm^2 + 5m - v = 0
m^2 = x \Rightarrow m = \sqrt{x} \\
x^2\sqrt{x} - 5vx^2 - 10x\sqrt{x} + 10vx + 5\sqrt{x} - v = 0
x(x^2 - 10x + 5)^2 = v^2(5x^2 - 10x + 1)^2
x^5 + (- 20 - 25v^2)x^4 + (110 + 100v^2) x^3 + (- 100 - 110v^2) x^2 + (25 + 20v^2) x - v^2 = 0

I know the five roots of this quintic.
v = \tan \theta
x = \tan^2 ((|\theta| + k \pi)/5)

The question now is:
I want to suppose that it is solvable by radicals (then what?)
calculate its discriminant somehow, like Cardano's method,
and explicit x = f(v).

Somebody might go to seventh degree. LOL

sweatfully

Vinicius
Vinicius Claudino Ferraz
2020-11-08 16:53:39 UTC
2² + 11² = 5³
10² + 55² = 5⁵
25² + 50² = 5⁵
38² + 41² = 5⁵
4² + 4² = 2⁵

The paper has a next page.

https://pbs.twimg.com/media/EmUMhr0XcAIFJPM?format=png&name=small

I used to know how-to solve by radicals. Why did I forget?

All we need is automation. Give me a faster processor! ; - D

procedure search for ugly numbers(degree: integer);
begin
if degree < 5 then exit;
if not is prime(degree) then exit;
...
Post by Vinicius Claudino Ferraz
sweatfully
Vinicius
Vinicius Claudino Ferraz
2020-11-09 15:27:26 UTC
This is a verification in R++ Language that the trisection formula works for complex variables.

Post by Vinicius Claudino Ferraz
sweatfully
Vinicius
Vinicius Claudino Ferraz
2020-11-11 15:15:42 UTC
The fifth degree holds!

https://www.wolframalpha.com/input/?i=tan+%285+*arccos+%288%2F120%5E5%29%29+-+%2840948974528705331149208412160000000007+sqrt%28197440574693877551%29%29%2F29249267520503807987906764800000000001

x^5 + (- 20 - 25v^2)x^4 + (110 + 100v^2) x^3 + (- 100 - 110v^2) x^2 + (25 + 20v^2) x - v^2 = 0 \\
s = - coef(x^4)/5 = 4 + 5v^2
x = y + s
(y + s)^5 - 5s(y + s)^4 + a_3 (y + s)^3 + a_2 (y + s)^2 + a_1 (y + s) + a_0 = 0
10y^3 s^2 - 5s * 4y^3 s + a_3 y^3 => coef(y^3) = -10 (4 + 5v^2)^2 + 110 + 100v^2

rad2 = [coef(y^3)/5]^5 = 2^5 [- (4 + 5v^2)^2 + 11 + 10v^2]^5

v = 1
b/a = tan 5 arccos (9 - m^2)/120^5
m = 1 => we have b/a => we have varphi = arctan b/a, it suffices to sepparate a from b.

a = r cos varphi
b = r sin varphi

Now I gotta go to n = 7.

x_7 + a_6 x_6 + ... = 0
s = - coef(x^6)/7
x = y + s
(y + s)^7 - 7s(y + s)^6 + a_5 (y + s)^5 + ... + a_2 (y + s)^2 + a_1 (y + s) + a_0 = 0
Newton(7, 2) y^5 s^2 - 7s * 6 y^5 s + a_5 y^5

For n = 7, we need a_6 and a_5.
For n = p prime, we need a_{p - 1} and a_{p - 2} from the formula <=== tan² pθ = f(tan² θ).

Vinicius
11/11/20/20
Vinicius Claudino Ferraz
2020-11-12 21:37:38 UTC
Finally, I've got the generalization.

https://pbs.twimg.com/media/Emp0h7SWMAYGTrR?format=jpg&name=large

Verified for p = 3 ==> Newton(3, 4) = 0

Verified for p = 5
https://t.co/sltbPPyy9R?amp=1

Verified for p = 7

Verified for p = 11
https://t.co/0Pgy2Upj6K?amp=1

Vinicius
12th/november/20/20
Vinicius Claudino Ferraz
2020-11-13 10:51:32 UTC
88^2 + 16^2 = 20^3

9736837^2 + 3545163^2 = 640^5
9793333^2 + 3385973^2 = 640^5
10105209^2 + 2293237^2 = 640^5
10354688^2 + 393216^2 = 640^5

6897954^2 + 8111605^2 = 220^6
5285438^2 + 9243595^2 = 220^6
3579299^2 + 10028386^2 = 220^6

738478566^2 + 1375152984^2 = 1160^6
306801311^2 + 1530447411^2 = 1160^6
554731364^2 + 1458996037^2 = 1160^6

4549053^2 + 14155362^2 = 112^7
13019854^2 + 7179940^2 = 112^7
9816880^2 + 11166781^2 = 112^7
5296776^2 + 13892887^2 = 112^7
6176768^2 + 13524632^2 = 112^7
9824500^2 + 11160078^2 = 112^7

600125323^2 + 697503298^2 = 364^7
856422095^2 + 336455717^2 = 364^7
714413204^2 + 579892255^2 = 364^7
614912876^2 + 684502307^2 = 364^7
650317224^2 + 650959878^2 = 364^7
722023296^2 + 570389002^2 = 364^7

2.43523e+11^2 + 177226896108^2 = 1904^7
279767334912^2 + 111548401290^2 = 1904^7
299974939044^2 + 26978574087^2 = 1904^7
244600905949^2 + 175736178184^2 = 1904^7
249955391581^2 + 168033061431^2 = 1904^7
255415309793^2 + 159611487798^2 = 1904^7

83146724721^2 + 52520027853^2 = 560^8
66743325956^2 + 72229215680^2 = 560^8
76384992206^2 + 61944040254^2 = 560^8
83146724721^2 + 52520027853^2 = 560^8
66743325956^2 + 72229215680^2 = 560^8
76384992206^2 + 61944040254^2 = 560^8

4.946286e+13^2 + 5.219106e+13^2 = 2912^8

Precision problem.
I need Python.
Peter
2020-11-03 14:14:07 UTC
Post by Vinicius Claudino Ferraz
https://pbs.twimg.com/media/El0Z7KvWoAUAqU5?format=png&name=900x900
\begin{align}
\tan^2 \biggl(\cfrac{\pi}{3} + \cfrac{1}{3} \text{ arctan } |x|\biggl) = 2 + 3x^2 + 2 \sqrt{x^2 + 1}\sqrt{9 x^2 + 1} \cos\biggl(\cfrac{1}{3} \text{ arctan } \cfrac{8|x|}{27 x^4 + 18 x^2 - 1}\biggl)
\end{align}
Use \tan and \arctan else they come out in different fonts.
Post by Vinicius Claudino Ferraz
m = tan θ
v = tan 3θ = (3m - m³)/(1 - 3m²)
v/m =1/3 - 8/(9m² - 3)
x = m²
x³ + px + q = 0
Cardano
--
When, once, reference was made to a statesman almost universally
recognized as one of the villains of this century, in order to
induce him to a negative judgment, he replied: "My situation is
so different from his, that it is not for me to pass judgment".
Ernst Specker on Paul Bernays
mitchr...@gmail.com
2020-11-13 19:14:00 UTC
Post by Vinicius Claudino Ferraz
https://pbs.twimg.com/media/El0Z7KvWoAUAqU5?format=png&name=900x900
\begin{align}
\tan^2 \biggl(\cfrac{\pi}{3} + \cfrac{1}{3} \text{ arctan } |x|\biggl) = 2 + 3x^2 + 2 \sqrt{x^2 + 1}\sqrt{9 x^2 + 1} \cos\biggl(\cfrac{1}{3} \text{ arctan } \cfrac{8|x|}{27 x^4 + 18 x^2 - 1}\biggl)
\end{align}
m = tan θ
v = tan 3θ = (3m - m³)/(1 - 3m²)
v/m =1/3 - 8/(9m² - 3)
x = m²
x³ + px + q = 0
Cardano
Zero in the trig table can't be a triangle.
No polygon can go to a zero angle
and remain a polygon...
Vinicius Claudino Ferraz
2020-11-13 20:41:58 UTC
Post by ***@gmail.com
Zero in the trig table can't be a triangle.
No polygon can go to a zero angle
and remain a polygon...
I agree. But a triangle may be the cubic root of the angle = zero.

z^n = 1 (cos 0 + i sin 0)

0 = 0 + 2 k pi

z = cos 2 k pi/n + i sin ~

0 <= k <= n - 1

It's a polygon of n sides. Don't you agree?

Do you believe today is my hello world in Python?
I don't know why windows calculator shows integer, R++ shows integer, but Python shows integer + 0.4.
We have a problem.

Out of charity, there is no salvation at all.

Vinicius
mitchr...@gmail.com
2020-11-14 02:51:47 UTC
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
Zero in the trig table can't be a triangle.
No polygon can go to a zero angle
and remain a polygon...
I agree. But a triangle may be the cubic root of the angle = zero.
You are making things up you moron...
Vinicius Claudino Ferraz
2020-11-14 23:14:05 UTC
Post by ***@gmail.com
You are making things up you moron...
This is a sample. This is also a correction. Several things are corrected.

https://pbs.twimg.com/media/Em0cBGoXcAknHjv?format=jpg&name=medium

With my Python's Lesson 2 = second day, I discovered that

360150403588837119 + 486510850526179585 = 364^7
733458804863979016 + 113202449251037688 = 364^7
510386226351227899 + 336275027763788805 = 364^7
378117845233971187 + 468543408881045517 = 364^7
422912491224802069 + 423748762890214635 = 364^7
521317639966703616 + 325343614148313088 = 364^7

And when I say that A + B = c^p

it is equivalent to construct z^(1/p) = c + di with ruler and compass

where z := sqrt(A) + i * sqrt(B)

Initially I've noted that
tan 2t = 2 tan t / (1 - tan^2 t)
v = 2m / (1 - m^2)
v - vm^2 = 2m
vm^2 + 2m + v = 0
this is the Newton's Triangle (1, 2, 1)

m^3 - 3vm^2 - 3m + v = 0

m^5 - 5vm^4 - 10m^3 + 10vm^2 + 5m - v = 0

m^7 - 7vm^6 - 21m^5 + 35vm^4 + 35m^3 - 21vm^2 - 7m + v = 0

I would be too fool if I did not know how to generalize. ; - )

whenever p >= 4, the factorials may be reduced to:

b = sqrt{p - 1}/sqrt{6} *
sqrt{ 9 p^2 v^4
+ (16p^2 - 14p) v^2
+ 7p^2 - 14p + 6 }

Vinicius
Nov/14th/2,020
Vinicius Claudino Ferraz
2020-11-15 18:33:30 UTC
Parla! Zero is equal to each one of these lines.

https://www.wolframalpha.com/input/?i=tan%2811*arccos%28%28tan%5E2%2845%2F11*pi%2F180%29-63%2F3%29%2Fsqrt%286160%29%29%29-sqrt%28112502659859281431343038111448362269%2F3036160521213918411432288551637731%29

tan(3*arccos((tan^2(45/3*pi/180)-15/3)/sqrt(80)))-sqrt(256/7744)
tan(3*arccos((tan^2(765/3*pi/180)-15/3)/sqrt(80)))-sqrt(256/7744)
tan(3*arccos((tan^2(60/3*pi/180)-33/3)/sqrt(448)))-sqrt(3072/1401856)
tan(3*arccos((tan^2(780/3*pi/180)-33/3)/sqrt(448)))-sqrt(3072/1401856)
tan(4*arccos((tan^2(420/4*pi/180)-45/3)/sqrt(1216)))-sqrt(128485023/8412232033)
tan(4*arccos((tan^2(1140/4*pi/180)-45/3)/sqrt(1216)))-sqrt(128485023/8412232033)
tan(5*arccos((tan^2(45/5*pi/180)-27/3)/sqrt(480)))-sqrt(6567397530/18315802470)
tan(5*arccos((tan^2(765/5*pi/180)-27/3)/sqrt(480)))-sqrt(5326861951/19556338049)
tan(5*arccos((tan^2(1125/5*pi/180)-27/3)/sqrt(480)))-sqrt(2147385344/22735814656)
tan(5*arccos((tan^2(60/5*pi/180)-57/3)/sqrt(2560)))-sqrt(12568182527263/94805999872737)
tan(5*arccos((tan^2(780/5*pi/180)-57/3)/sqrt(2560)))-sqrt(11464811733824/95909370666176)
tan(5*arccos((tan^2(1140/5*pi/180)-57/3)/sqrt(2560)))-sqrt(5258936654114/102115245745886)
tan(5*arccos((tan^2(1500/5*pi/180)-57/3)/sqrt(2560)))-sqrt(154618822656/107219563577344)
tan(6*arccos((tan^2(405/6*pi/180)-33/3)/sqrt(880)))-sqrt(85444049565645/27935854434355)
tan(6*arccos((tan^2(1485/6*pi/180)-33/3)/sqrt(880)))-sqrt(85444049565645/27935854434355)
tan(6*arccos((tan^2(420/6*pi/180)-69/3)/sqrt(4640)))-sqrt(2342269278338771208/94127044477228792)
tan(6*arccos((tan^2(1500/6*pi/180)-69/3)/sqrt(4640)))-sqrt(2342269278338771208/94127044477228792)
tan(7*arccos((tan^2(30/7*pi/180)-25/3)/sqrt(448)))-sqrt(200374260411389/20693880329219)
tan(7*arccos((tan^2(390/7*pi/180)-25/3)/sqrt(448)))-sqrt(51551540984637/169516599755971)
tan(7*arccos((tan^2(750/7*pi/180)-25/3)/sqrt(448)))-sqrt(124697007803733/96371132936875)
tan(7*arccos((tan^2(1110/7*pi/180)-25/3)/sqrt(448)))-sqrt(193012309371266/28055831369342)
tan(7*arccos((tan^2(1470/7*pi/180)-25/3)/sqrt(448)))-sqrt(182915677814784/38152462925824)
tan(7*arccos((tan^2(2190/7*pi/180)-25/3)/sqrt(448)))-sqrt(124547349335769/96520791404839)
tan(7*arccos((tan^2(45/7*pi/180)-39/3)/sqrt(1456)))-sqrt(486510850526179585/360150403588837119)
tan(7*arccos((tan^2(405/7*pi/180)-39/3)/sqrt(1456)))-sqrt(113202449251037688/733458804863979016)
tan(7*arccos((tan^2(1125/7*pi/180)-39/3)/sqrt(1456)))-sqrt(468543408881045517/378117845233971187)
tan(7*arccos((tan^2(1485/7*pi/180)-39/3)/sqrt(1456)))-sqrt(423748762890214635/422912491224802069)
tan(7*arccos((tan^2(2205/7*pi/180)-39/3)/sqrt(1456)))-sqrt(325343614148313088/521317639966703616)
tan(7*arccos((tan^2(60/7*pi/180)-81/3)/sqrt(7616)))-sqrt(31409372704065354115026/59303434810012969818158)
tan(7*arccos((tan^2(420/7*pi/180)-81/3)/sqrt(7616)))-sqrt(12443045830315149885440/78269761683763174047744)
tan(7*arccos((tan^2(780/7*pi/180)-81/3)/sqrt(7616)))-sqrt(727843459783315390404/89984964054295008542780)
tan(7*arccos((tan^2(1140/7*pi/180)-81/3)/sqrt(7616)))-sqrt(30883204322839290103213/59829603191239033829971)
tan(7*arccos((tan^2(1500/7*pi/180)-81/3)/sqrt(7616)))-sqrt(28235109733830560760256/62477697780247763172928)
tan(7*arccos((tan^2(2220/7*pi/180)-81/3)/sqrt(7616)))-sqrt(25475827037246891841866/65236980476831432091318)
tan(8*arccos((tan^2(780/8*pi/180)-93/3)/sqrt(11648)))-sqrt(4276806604555147741561452034/893674616483995025966795262)
tan(8*arccos((tan^2(2220/8*pi/180)-93/3)/sqrt(11648)))-sqrt(4276806604555147741561452034/893674616483995025966795262)
tan(9*arccos((tan^2(390/9*pi/180)-33/3)/sqrt(1024)))-sqrt(4436430680862688415384/285935802006956798312)
tan(9*arccos((tan^2(1110/9*pi/180)-33/3)/sqrt(1024)))-sqrt(2915355539789133888947/1807010943080511324749)
tan(9*arccos((tan^2(1470/9*pi/180)-33/3)/sqrt(1024)))-sqrt(4721869791356346078212/496691513299135484)
tan(9*arccos((tan^2(1830/9*pi/180)-33/3)/sqrt(1024)))-sqrt(4715153515648555489803/7212967221089723893)
tan(9*arccos((tan^2(2190/9*pi/180)-33/3)/sqrt(1024)))-sqrt(799897308916906858547/3922469173952738355149)
tan(9*arccos((tan^2(2910/9*pi/180)-33/3)/sqrt(1024)))-sqrt(4618472662981243744574/103893819888401469122)
tan(9*arccos((tan^2(45/9*pi/180)-51/3)/sqrt(3264)))-sqrt(133296821558481397655045656/27105787751689263063202280)
tan(9*arccos((tan^2(405/9*pi/180)-51/3)/sqrt(3264)))-sqrt(111208107163347894193356800/49194502146822766524891136)
tan(9*arccos((tan^2(1125/9*pi/180)-51/3)/sqrt(3264)))-sqrt(84744545562388512792227984/75658063747782147926019952)
tan(9*arccos((tan^2(1485/9*pi/180)-51/3)/sqrt(3264)))-sqrt(132012938197412684337318903/28389671112757976380929033)
tan(9*arccos((tan^2(1845/9*pi/180)-51/3)/sqrt(3264)))-sqrt(129014248674584791690468577/31388360635585869027779359)
tan(9*arccos((tan^2(2205/9*pi/180)-51/3)/sqrt(3264)))-sqrt(24145460318779590348242365/136257148991391070370005571)
tan(9*arccos((tan^2(2925/9*pi/180)-51/3)/sqrt(3264)))-sqrt(123106433224711163953697460/37296176085459496764550476)
tan(9*arccos((tan^2(60/9*pi/180)-105/3)/sqrt(16896)))-sqrt(255075030826108768499562183169753/172995356595786129566904589183271)
tan(9*arccos((tan^2(420/9*pi/180)-105/3)/sqrt(16896)))-sqrt(221217015039313469574841675712594/206853372382581428491625096640430)
tan(9*arccos((tan^2(1140/9*pi/180)-105/3)/sqrt(16896)))-sqrt(200342077137829142231147702681025/227728310284065755835319069671999)
tan(9*arccos((tan^2(1500/9*pi/180)-105/3)/sqrt(16896)))-sqrt(253790387084013129704637142321727/174280000337881768361829630031297)
tan(9*arccos((tan^2(1860/9*pi/180)-105/3)/sqrt(16896)))-sqrt(247848736900803631130797652766671/180221650521091266935669119586353)
tan(9*arccos((tan^2(2220/9*pi/180)-105/3)/sqrt(16896)))-sqrt(97902644426114795129611360543546/330167742995780102936855411809478)
tan(9*arccos((tan^2(2580/9*pi/180)-105/3)/sqrt(16896)))-sqrt(3416610519681090210736914318519/424653776902213807855729858034505)
tan(9*arccos((tan^2(2940/9*pi/180)-105/3)/sqrt(16896)))-sqrt(242361172297413289825694428475219/185709215124481608240772343877805)
tan(10*arccos((tan^2(30/10*pi/180)-37/3)/sqrt(1440)))-sqrt(1020854370495893727781704/35540730030133866272218296)
tan(10*arccos((tan^2(1110/10*pi/180)-37/3)/sqrt(1440)))-sqrt(36168926768443004069842035/392657632186755930157965)
tan(10*arccos((tan^2(1470/10*pi/180)-37/3)/sqrt(1440)))-sqrt(96389989908676952360214/36465194410721083047639786)
tan(10*arccos((tan^2(1830/10*pi/180)-37/3)/sqrt(1440)))-sqrt(1020854370495893727781704/35540730030133866272218296)
tan(10*arccos((tan^2(2910/10*pi/180)-37/3)/sqrt(1440)))-sqrt(36168926768443004069842035/392657632186755930157965)
tan(10*arccos((tan^2(3270/10*pi/180)-37/3)/sqrt(1440)))-sqrt(96389989908676952360214/36465194410721083047639786)
tan(10*arccos((tan^2(765/10*pi/180)-57/3)/sqrt(4560)))-sqrt(217079757722854997225864052069/3490141556395711392534135947931)
tan(10*arccos((tan^2(2565/10*pi/180)-57/3)/sqrt(4560)))-sqrt(217079757722854997225864052069/3490141556395711392534135947931)
tan(10*arccos((tan^2(780/10*pi/180)-117/3)/sqrt(23520)))-sqrt(39320231951652944839877452308815185057/10085036194423477633627069931184814943)
tan(10*arccos((tan^2(2580/10*pi/180)-117/3)/sqrt(23520)))-sqrt(39320231951652944839877452308815185057/10085036194423477633627069931184814943)
tan(11*arccos((tan^2(45/11*pi/180)-63/3)/sqrt(6160)))-sqrt(112502659859281431343038111448362269/3036160521213918411432288551637731)
tan(11*arccos((tan^2(405/11*pi/180)-63/3)/sqrt(6160)))-sqrt(108822094165051988985109487493321726/6716726215443360769360912506678274)
tan(11*arccos((tan^2(765/11*pi/180)-63/3)/sqrt(6160)))-sqrt(15506870759256197277194695217004174/100031949621239152477275704782995826)
tan(11*arccos((tan^2(1125/11*pi/180)-63/3)/sqrt(6160)))-sqrt(115499444285357358447692135363723622/39376095137991306778264636276378)
tan(11*arccos((tan^2(1485/11*pi/180)-63/3)/sqrt(6160)))-sqrt(104968947132267318543974400000000000/10569873248228031210496000000000000)
tan(11*arccos((tan^2(1845/11*pi/180)-63/3)/sqrt(6160)))-sqrt(112271633479366433675729880649188818/3267186901128916078740519350811182)
tan(11*arccos((tan^2(2205/11*pi/180)-63/3)/sqrt(6160)))-sqrt(111741014794806685053024920134228453/3797805585688664701445479865771547)
tan(11*arccos((tan^2(2565/11*pi/180)-63/3)/sqrt(6160)))-sqrt(96257509111315940992269158023467161/19281311269179408762201241976532839)
tan(11*arccos((tan^2(3285/11*pi/180)-63/3)/sqrt(6160)))-sqrt(73543792326841564641136799451783554/41995028053653785113333600548216446)
tan(11*arccos((tan^2(3645/11*pi/180)-63/3)/sqrt(6160)))-sqrt(110731299677991589259189226506190748/4807520702503760495281173493809252)
tan(11*arccos((tan^2(60/11*pi/180)-129/3)/sqrt(31680)))-sqrt(6186413571688538926721035847230711988232460/1504483470109300782647763404430088011767540)
tan(11*arccos((tan^2(420/11*pi/180)-129/3)/sqrt(31680)))-sqrt(5942935002041064574720316735087575586497953/1747962039756775134648482516573224413502047)
tan(11*arccos((tan^2(780/11*pi/180)-129/3)/sqrt(31680)))-sqrt(2341685161518844999257015368952340137162123/5349211880278994710111783882708459862837877)
tan(11*arccos((tan^2(1140/11*pi/180)-129/3)/sqrt(31680)))-sqrt(13790374653997086331468450309648183355591/7677106667143842623037330801351151816644409)
tan(11*arccos((tan^2(1500/11*pi/180)-129/3)/sqrt(31680)))-sqrt(5822305259707846003565976699610683468622414/1868591782089993705802822552050116531377586)
tan(11*arccos((tan^2(1860/11*pi/180)-129/3)/sqrt(31680)))-sqrt(6175507211381886249902717454208111898134053/1515389830415953459466081797452688101865947)
tan(11*arccos((tan^2(2220/11*pi/180)-129/3)/sqrt(31680)))-sqrt(6127252348474023078159353196065782376127687/1563644693323816631209446055595017623872313)
tan(11*arccos((tan^2(2580/11*pi/180)-129/3)/sqrt(31680)))-sqrt(5359980708016164459144056661793646695497946/2330916333781675250224742589867153304502054)
tan(11*arccos((tan^2(3300/11*pi/180)-129/3)/sqrt(31680)))-sqrt(4887210314475127101659686948044800000000000/2803686727322712607709112303616000000000000)
tan(11*arccos((tan^2(3660/11*pi/180)-129/3)/sqrt(31680)))-sqrt(6085463092929513318150549215290519285393341/1605433948868326391218250036370280714606659)
tan(12*arccos((tan^2(1140/12*pi/180)-141/3)/sqrt(41536)))-sqrt(1373926524955042943615276507029349987090637956887/197796999659800684671655933428375392715758196969)
tan(12*arccos((tan^2(3300/12*pi/180)-141/3)/sqrt(41536)))-sqrt(1373926524955042943615276507029349987090637956887/197796999659800684671655933428375392715758196969)
tan(13*arccos((tan^2(1470/13*pi/180)-49/3)/sqrt(3328)))-sqrt(54648779620395923954919322784563030634/36889604499080984516092814080915182998)
tan(13*arccos((tan^2(3270/13*pi/180)-49/3)/sqrt(3328)))-sqrt(781732622804332150563873789490126608/90756651496672576320448263075988087024)
tan(13*arccos((tan^2(765/13*pi/180)-75/3)/sqrt(10400)))-sqrt(229102039396321614773980593050813608720302918/19013247924052042826019406949186391279697082)
tan(13*arccos((tan^2(1485/13*pi/180)-75/3)/sqrt(10400)))-sqrt(176483945313290822833426018198189298370229233/71631342007082834766573981801810701629770767)
tan(13*arccos((tan^2(1845/13*pi/180)-75/3)/sqrt(10400)))-sqrt(248113694544933330609376926165905497953310900/1592775440326990623073834094502046689100)
tan(13*arccos((tan^2(2925/13*pi/180)-75/3)/sqrt(10400)))-sqrt(247413400646922480099951970532671855479750656/701886673451177500048029467328144520249344)
tan(13*arccos((tan^2(3285/13*pi/180)-75/3)/sqrt(10400)))-sqrt(23068104477106295431654631567843040919773469/225047182843267362168345368432156959080226531)
tan(13*arccos((tan^2(3645/13*pi/180)-75/3)/sqrt(10400)))-sqrt(166884419222754953786624251153248516186909886/81230868097618703813375748846751483813090114)
tan(13*arccos((tan^2(4005/13*pi/180)-75/3)/sqrt(10400)))-sqrt(243575913970768438044467848162150845488254909/4539373349605219555532151837849154511745091)
tan(13*arccos((tan^2(60/13*pi/180)-153/3)/sqrt(53248)))-sqrt(388022232272158473847172132479924939746186626046777515/24230000338415059740720984100791805003627105455023957)
tan(13*arccos((tan^2(420/13*pi/180)-153/3)/sqrt(53248)))-sqrt(383430394666661631734907407084614440812966250656542522/28821837943911901852985709496102303936847480845258950)
tan(13*arccos((tan^2(780/13*pi/180)-153/3)/sqrt(53248)))-sqrt(344461681645264255432833324686191591016217496545918976/67790550965309278155059791894525153733596234955882496)
tan(13*arccos((tan^2(1500/13*pi/180)-153/3)/sqrt(53248)))-sqrt(317322890310454504717624615027781157997473837137172001/94929342300119028870268501552935586752339894364629471)
tan(13*arccos((tan^2(1860/13*pi/180)-153/3)/sqrt(53248)))-sqrt(381397896618823966271342166862733755079216202383012533/30854335991749567316550949717982989670597529118788939)
tan(13*arccos((tan^2(2220/13*pi/180)-153/3)/sqrt(53248)))-sqrt(387798896456549727830080486465280828161401353716856179/24453336154023805757812630115435916588412377784945293)
tan(13*arccos((tan^2(2580/13*pi/180)-153/3)/sqrt(53248)))-sqrt(386833432059689036927694934584861739466456933096987942/25418800550884496660198181995855005283356798404813530)
tan(13*arccos((tan^2(2940/13*pi/180)-153/3)/sqrt(53248)))-sqrt(374585037062146236072677905964325737187368302241169075/37667195548427297515215210616391007562445429260632397)
tan(13*arccos((tan^2(3300/13*pi/180)-153/3)/sqrt(53248)))-sqrt(147642751859124637717069427674490528536024859880699696/264609480751448895870823688906226216213788871621101776)
tan(13*arccos((tan^2(4020/13*pi/180)-153/3)/sqrt(53248)))-sqrt(368689219032067211087605613034782728574258794620016698/43563013578506322500287503545934016175554936881784774)
tan(13*arccos((tan^2(4380/13*pi/180)-153/3)/sqrt(53248)))-sqrt(386024887951399066393884355041833951275012676709562739/26227344659174467194008761538882793474801054792238733)
tan(14*arccos((tan^2(45/14*pi/180)-81/3)/sqrt(13104)))-sqrt(594176034155089475076060836600990553624651098257/15804715872946012391970274453280561376757204260719)
tan(14*arccos((tan^2(405/14*pi/180)-81/3)/sqrt(13104)))-sqrt(383297042938726743384175946603148174052036551952/16015594864162375123662159343278403756329818807024)
tan(14*arccos((tan^2(1845/14*pi/180)-81/3)/sqrt(13104)))-sqrt(19556709952364467235440016515709980675330496994/16379335197148737399810895273365841949706524861982)
tan(14*arccos((tan^2(2205/14*pi/180)-81/3)/sqrt(13104)))-sqrt(471163897055765457309414120348951547710818855703/15927728010045336409736921169532600382671036503273)
tan(14*arccos((tan^2(2565/14*pi/180)-81/3)/sqrt(13104)))-sqrt(594176034155089475076060836600990553624651098257/15804715872946012391970274453280561376757204260719)
tan(14*arccos((tan^2(2925/14*pi/180)-81/3)/sqrt(13104)))-sqrt(383297042938726743384175946603148174052036551952/16015594864162375123662159343278403756329818807024)
tan(14*arccos((tan^2(4365/14*pi/180)-81/3)/sqrt(13104)))-sqrt(19556709952364467235440016515709980675330496994/16379335197148737399810895273365841949706524861982)
tan(14*arccos((tan^2(4725/14*pi/180)-81/3)/sqrt(13104)))-sqrt(471163897055765457309414120348951547710818855703/15927728010045336409736921169532600382671036503273)
tan(14*arccos((tan^2(1140/14*pi/180)-165/3)/sqrt(66976)))-sqrt(42686128983764569442031601644519327306364729981500971219286/93467435195044353499141612819032302949754733710483750192810)
tan(14*arccos((tan^2(3660/14*pi/180)-165/3)/sqrt(66976)))-sqrt(42686128983764569442031601644519327306364729981500971219286/93467435195044353499141612819032302949754733710483750192810)
tan(15*arccos((tan^2(1125/15*pi/180)-87/3)/sqrt(16240)))-sqrt(56923723438501851994903005460563410213592288226592904/1285502736234250493555690838062394950362407711773407096)
tan(15*arccos((tan^2(1485/15*pi/180)-87/3)/sqrt(16240)))-sqrt(110172942486103971746129594025205522295504504550087765/1232253517186648373804464249497752838280495495449912235)
tan(15*arccos((tan^2(3645/15*pi/180)-87/3)/sqrt(16240)))-sqrt(1307524844653276171592628775865699727045885341457328764/34901615019476173957965067657258633530114658542671236)
tan(15*arccos((tan^2(4365/15*pi/180)-87/3)/sqrt(16240)))-sqrt(1018484136168079826603825179566244661619713410470740014/323942323504672518946768663956713698956286589529259986)
tan(15*arccos((tan^2(60/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55582029462437971503736988999561740181851254543615788675981669630/116825359739263661544053232446390794175853636352211324018330370)
tan(15*arccos((tan^2(420/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55494546078033344041660915274501686023111750940780219313386356416/204308744143891123620126957506444952915357239187780686613643584)
tan(15*arccos((tan^2(780/15*pi/180)-177/3)/sqrt(82880)))-sqrt(54723625738926775939598102604043324318773371906003378298016305787/975229083250459225682939627964806657253736273964621701983694213)
tan(15*arccos((tan^2(1140/15*pi/180)-177/3)/sqrt(82880)))-sqrt(21669593398810095671091484855071161887366234036912708634249811778/34029261423367139494189557376936969088660874143055291365750188222)
tan(15*arccos((tan^2(1860/15*pi/180)-177/3)/sqrt(82880)))-sqrt(54241569333334944943540388655195612924173191318923608450252581548/1457285488842290221740653576812518051853916861044391549747418452)
tan(15*arccos((tan^2(2220/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55454111895770365481399136901561018714465595569659707521627313641/244742926406869683881905330447112261561512610308292478372686359)
tan(15*arccos((tan^2(2580/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55577962735803070548163237342504431794334186403658709928694413656/120892086374164617117804889503699181692921776309290071305586344)
tan(15*arccos((tan^2(2940/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55560109990716166298643349621505607382933291184039390806256972396/138744831461068866637692610502523593093816995928609193743027604)
tan(15*arccos((tan^2(3300/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55315797254722058773877709508148582498766787419993912364698205804/383057567455176391403332723859548477260320759974087635301794196)
tan(15*arccos((tan^2(3660/15*pi/180)-177/3)/sqrt(82880)))-sqrt(51764915371678461812917600726632308564084671136516996959427596124/3933939450498773352363441505375822411942437043451003040572403876)
tan(15*arccos((tan^2(4380/15*pi/180)-177/3)/sqrt(82880)))-sqrt(48379617355525155913016103695043128855590347562456180776207587320/7319237466652079252264938536965002120436760617511819223792412680)
tan(15*arccos((tan^2(4740/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55195792844642197019288042184993662976915318179074894871267302684/503061977535038145993000047014467999111790000893105128732697316)
tan(15*arccos((tan^2(5100/15*pi/180)-177/3)/sqrt(82880)))-sqrt(55544860426650102960008292267893793519726059611076863353871758488/153994395527132205272749964114337456301048568891136646128241512)
tan(16*arccos((tan^2(30/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1255117208603206189190576959000134579516294882415576/589557198767748972409423040999865420483705117584424)
tan(16*arccos((tan^2(390/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1181387538273882139317821176016587009044753808982334/663286869097073022282178823983412990955246191017666)
tan(16*arccos((tan^2(750/16*pi/180)-61/3)/sqrt(6400)))-sqrt(829334970107006102461751857769141501694307397855688/1015339437263949059138248142230858498305692602144312)
tan(16*arccos((tan^2(2190/16*pi/180)-61/3)/sqrt(6400)))-sqrt(929067421591499785002540681862570902125237286728096/915606985779455376597459318137429097874762713271904)
tan(16*arccos((tan^2(2550/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1204579689468689880968881596812477119069008349970909/640094717902265280631118403187522880930991650029091)
tan(16*arccos((tan^2(2910/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1255117208603206189190576959000134579516294882415576/589557198767748972409423040999865420483705117584424)
tan(16*arccos((tan^2(3270/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1181387538273882139317821176016587009044753808982334/663286869097073022282178823983412990955246191017666)
tan(16*arccos((tan^2(3630/16*pi/180)-61/3)/sqrt(6400)))-sqrt(829334970107006102461751857769141501694307397855688/1015339437263949059138248142230858498305692602144312)
tan(16*arccos((tan^2(5070/16*pi/180)-61/3)/sqrt(6400)))-sqrt(929067421591499785002540681862570902125237286728096/915606985779455376597459318137429097874762713271904)
tan(16*arccos((tan^2(5430/16*pi/180)-61/3)/sqrt(6400)))-sqrt(1204579689468689880968881596812477119069008349970909/640094717902265280631118403187522880930991650029091)
tan(16*arccos((tan^2(45/16*pi/180)-93/3)/sqrt(19840)))-sqrt(21179182184359172513497953796930208942013079066225311778586/113006699417332741701626578430509771461739880933774688221414)
tan(16*arccos((tan^2(405/16*pi/180)-93/3)/sqrt(19840)))-sqrt(18720462381368870028562489600290166850969053160969045606400/115465419220323044186562042627149813552783906839030954393600)
tan(16*arccos((tan^2(765/16*pi/180)-93/3)/sqrt(19840)))-sqrt(9355848236574733303181933697038609028460044110004350080461/124830033365117180911942598530401371375292915889995649919539)
tan(16*arccos((tan^2(2205/16*pi/180)-93/3)/sqrt(19840)))-sqrt(12741412828285077207874202192079983064008896605053888092757/121444468773406837007250330035359997339744063394946111907243)
tan(16*arccos((tan^2(2565/16*pi/180)-93/3)/sqrt(19840)))-sqrt(19767993020265014324019734840494804273247636307433502749900/114417888581426899891104797386945176130505323692566497250100)
tan(16*arccos((tan^2(2925/16*pi/180)-93/3)/sqrt(19840)))-sqrt(21179182184359172513497953796930208942013079066225311778586/113006699417332741701626578430509771461739880933774688221414)
tan(16*arccos((tan^2(3285/16*pi/180)-93/3)/sqrt(19840)))-sqrt(18720462381368870028562489600290166850969053160969045606400/115465419220323044186562042627149813552783906839030954393600)
tan(16*arccos((tan^2(3645/16*pi/180)-93/3)/sqrt(19840)))-sqrt(9355848236574733303181933697038609028460044110004350080461/124830033365117180911942598530401371375292915889995649919539)
tan(16*arccos((tan^2(5085/16*pi/180)-93/3)/sqrt(19840)))-sqrt(12741412828285077207874202192079983064008896605053888092757/121444468773406837007250330035359997339744063394946111907243)
tan(16*arccos((tan^2(5445/16*pi/180)-93/3)/sqrt(19840)))-sqrt(19767993020265014324019734840494804273247636307433502749900/114417888581426899891104797386945176130505323692566497250100)
tan(16*arccos((tan^2(60/16*pi/180)-189/3)/sqrt(101120)))-sqrt(67224124002339599861186157421706307277950708889420103188703209924573/27757643941900897875977198759353216425305078502991068856811296790075427)
tan(16*arccos((tan^2(420/16*pi/180)-189/3)/sqrt(101120)))-sqrt(37890105827071065103330841900628280527846167322791741610328500301811/27786977960076166410735054074874294452055183044557697218389671499698189)
tan(16*arccos((tan^2(1500/16*pi/180)-189/3)/sqrt(101120)))-sqrt(4491096136304034012250488119078187194730428376968415003462645129397747/23333771929599203463587896797696735537852600834912073956537354870602253)
tan(16*arccos((tan^2(2220/16*pi/180)-189/3)/sqrt(101120)))-sqrt(2739783552770594353772682748339032065049171353471128360114855880812/27822128282350466881484612234026583700517980040527017831639885144119188)
tan(16*arccos((tan^2(2580/16*pi/180)-189/3)/sqrt(101120)))-sqrt(52565343622302813871217489584390766613680144928410533045490495834686/27772302722280934661967167427190531965969349066952078426954509504165314)
tan(16*arccos((tan^2(2940/16*pi/180)-189/3)/sqrt(101120)))-sqrt(67224124002339599861186157421706307277950708889420103188703209924573/27757643941900897875977198759353216425305078502991068856811296790075427)
tan(16*arccos((tan^2(3300/16*pi/180)-189/3)/sqrt(101120)))-sqrt(37890105827071065103330841900628280527846167322791741610328500301811/27786977960076166410735054074874294452055183044557697218389671499698189)
tan(16*arccos((tan^2(4380/16*pi/180)-189/3)/sqrt(101120)))-sqrt(4491096136304034012250488119078187194730428376968415003462645129397747/23333771929599203463587896797696735537852600834912073956537354870602253)
tan(16*arccos((tan^2(5100/16*pi/180)-189/3)/sqrt(101120)))-sqrt(2739783552770594353772682748339032065049171353471128360114855880812/27822128282350466881484612234026583700517980040527017831639885144119188)
tan(16*arccos((tan^2(5460/16*pi/180)-189/3)/sqrt(101120)))-sqrt(52565343622302813871217489584390766613680144928410533045490495834686/27772302722280934661967167427190531965969349066952078426954509504165314)
tan(17*arccos((tan^2(1125/17*pi/180)-99/3)/sqrt(23936)))-sqrt(16112746653116005697388204326595530925031255664801988624301856647/62724614177387745446161334609018233537823383271661456114149497)
tan(17*arccos((tan^2(1845/17*pi/180)-99/3)/sqrt(23936)))-sqrt(12688597436028518931365134278094341683793255854602267964190218773/3486873831264874511469231383110207474775823193471382116225787371)
tan(17*arccos((tan^2(4365/17*pi/180)-99/3)/sqrt(23936)))-sqrt(82509590320876642141847099276318130707751917424721703981933846/16092961676972516800692518561928231027861327130648928376434072298)
tan(17*arccos((tan^2(1140/17*pi/180)-201/3)/sqrt(121856)))-sqrt(16460770145155367494571799342560626514501607794628397103664743116967441017584/303233784868342284882444848921406555919000782635253565253019775057075688720)
tan(17*arccos((tan^2(1860/17*pi/180)-201/3)/sqrt(121856)))-sqrt(15680267670988062649117108566112338136501093100503055483920691316268730403369/1083736259035647130337135625369694933919515476760595184997071575755786302935)
tan(17*arccos((tan^2(4380/17*pi/180)-201/3)/sqrt(121856)))-sqrt(6682424753735986589956328937421082781649536330296042706199065634878352725065/10081579176287723189497915254060950288771072246967607962718697257146163981239)
tan(17*arccos((tan^2(5100/17*pi/180)-201/3)/sqrt(121856)))-sqrt(16763161887218306953493654568552027894789010146286920480971965703538249564160/842042805402825960589622930005175631598430976730187945797188486267142144)
tan(18*arccos((tan^2(30/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3107867153257895100750746263735457899975726060213699794802111/454846642427651649153974068200062234589563162903479295830593)
tan(18*arccos((tan^2(390/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3033540461768135391325811411543459463732209831194305009612392/529173333917411358578908920392060670833079391922874081020312)
tan(18*arccos((tan^2(750/18*pi/180)-69/3)/sqrt(9248)))-sqrt(2690081187598001318010188503049459052912849589280098351326240/872632608087545431894531828886061081652439633837080739306464)
tan(18*arccos((tan^2(1110/18*pi/180)-69/3)/sqrt(9248)))-sqrt(958639241759023666585744549993381911304415611412663404536157/2604074553926523083318975781942138223260873611704515686096547)
tan(18*arccos((tan^2(2190/18*pi/180)-69/3)/sqrt(9248)))-sqrt(1480587407969077257567375532649571111398909639367683118961687/2082126387716469492337344799285949023166379583749495971671017)
tan(18*arccos((tan^2(2550/18*pi/180)-69/3)/sqrt(9248)))-sqrt(2786469306510022138735147333593441050568982529109186020223829/776244489175524611169572998342079083996306694007993070408875)
tan(18*arccos((tan^2(2910/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3056775460565300007560774312684097571778982333707323409872820/505938335120246742343946019251422562786306889409855680759884)
tan(18*arccos((tan^2(3270/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3107867153257895100750746263735457899975726060213699794802111/454846642427651649153974068200062234589563162903479295830593)
tan(18*arccos((tan^2(3630/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3033540461768135391325811411543459463732209831194305009612392/529173333917411358578908920392060670833079391922874081020312)
tan(18*arccos((tan^2(3990/18*pi/180)-69/3)/sqrt(9248)))-sqrt(2690081187598001318010188503049459052912849589280098351326240/872632608087545431894531828886061081652439633837080739306464)
tan(18*arccos((tan^2(4350/18*pi/180)-69/3)/sqrt(9248)))-sqrt(958639241759023666585744549993381911304415611412663404536157/2604074553926523083318975781942138223260873611704515686096547)
tan(18*arccos((tan^2(5430/18*pi/180)-69/3)/sqrt(9248)))-sqrt(1480587407969077257567375532649571111398909639367683118961687/2082126387716469492337344799285949023166379583749495971671017)
tan(18*arccos((tan^2(5790/18*pi/180)-69/3)/sqrt(9248)))-sqrt(2786469306510022138735147333593441050568982529109186020223829/776244489175524611169572998342079083996306694007993070408875)
tan(18*arccos((tan^2(6150/18*pi/180)-69/3)/sqrt(9248)))-sqrt(3056775460565300007560774312684097571778982333707323409872820/505938335120246742343946019251422562786306889409855680759884)
tan(18*arccos((tan^2(45/18*pi/180)-105/3)/sqrt(28560)))-sqrt(770379891013167974813969538716060474651736894023590819555150009474682/1555395719704608504401538061023183431129087789029465180444849990525318)
tan(18*arccos((tan^2(405/18*pi/180)-105/3)/sqrt(28560)))-sqrt(730221231943508879549475810445237134650829807948533662605936602172661/1595554378774267599666031789294006771129994875104522337394063397827339)
tan(18*arccos((tan^2(765/18*pi/180)-105/3)/sqrt(28560)))-sqrt(578416067237116008846438879062973935977968375655747512422081526028569/1747359543480660470369068720676269969802856307397308487577918473971431)
tan(18*arccos((tan^2(1125/18*pi/180)-105/3)/sqrt(28560)))-sqrt(103565936787404256154403897330730450719704611616560859184174148977450/2222209673930372223061103702408513455061120071436495140815825851022550)
tan(18*arccos((tan^2(1485/18*pi/180)-105/3)/sqrt(28560)))-sqrt(1004199860989112997207229045435955631725662240665069466766442820304985/1321575749728663482008278554303288274055162442387986533233557179695015)
tan(18*arccos((tan^2(1845/18*pi/180)-105/3)/sqrt(28560)))-sqrt(2325623852182434540447997636334959797234376917667693992605840405743930/151758535341938767509963404284108546447765385362007394159594256070)
tan(18*arccos((tan^2(2205/18*pi/180)-105/3)/sqrt(28560)))-sqrt(266938872256256640705729679644877037734719178135520560744556325939910/2058836738461519838509777920094366868046105504917535439255443674060090)
tan(18*arccos((tan^2(2565/18*pi/180)-105/3)/sqrt(28560)))-sqrt(634126403791386329275831268705441484903030045206155104329179259789632/1691649206926390149939676331033802420877794637846900895670820740210368)
tan(18*arccos((tan^2(2925/18*pi/180)-105/3)/sqrt(28560)))-sqrt(747230750083991437390974208906790320993776781337095408314199262615377/1578544860633785041824533390832453584787047901715960591685800737384623)
tan(18*arccos((tan^2(3285/18*pi/180)-105/3)/sqrt(28560)))-sqrt(770379891013167974813969538716060474651736894023590819555150009474682/1555395719704608504401538061023183431129087789029465180444849990525318)
tan(18*arccos((tan^2(3645/18*pi/180)-105/3)/sqrt(28560)))-sqrt(730221231943508879549475810445237134650829807948533662605936602172661/1595554378774267599666031789294006771129994875104522337394063397827339)
tan(18*arccos((tan^2(4005/18*pi/180)-105/3)/sqrt(28560)))-sqrt(578416067237116008846438879062973935977968375655747512422081526028569/1747359543480660470369068720676269969802856307397308487577918473971431)
tan(18*arccos((tan^2(4365/18*pi/180)-105/3)/sqrt(28560)))-sqrt(103565936787404256154403897330730450719704611616560859184174148977450/2222209673930372223061103702408513455061120071436495140815825851022550)
tan(18*arccos((tan^2(4725/18*pi/180)-105/3)/sqrt(28560)))-sqrt(1004199860989112997207229045435955631725662240665069466766442820304985/1321575749728663482008278554303288274055162442387986533233557179695015)
tan(18*arccos((tan^2(5085/18*pi/180)-105/3)/sqrt(28560)))-sqrt(2325623852182434540447997636334959797234376917667693992605840405743930/151758535341938767509963404284108546447765385362007394159594256070)
tan(18*arccos((tan^2(5445/18*pi/180)-105/3)/sqrt(28560)))-sqrt(266938872256256640705729679644877037734719178135520560744556325939910/2058836738461519838509777920094366868046105504917535439255443674060090)
tan(18*arccos((tan^2(5805/18*pi/180)-105/3)/sqrt(28560)))-sqrt(634126403791386329275831268705441484903030045206155104329179259789632/1691649206926390149939676331033802420877794637846900895670820740210368)
tan(18*arccos((tan^2(6165/18*pi/180)-105/3)/sqrt(28560)))-sqrt(747230750083991437390974208906790320993776781337095408314199262615377/1578544860633785041824533390832453584787047901715960591685800737384623)
tan(18*arccos((tan^2(60/18*pi/180)-213/3)/sqrt(145248)))-sqrt(633007263962644773374830690915797168219519435947288814903078397266056348317814894/11414662515463331195932967645739158159264045058833545520175559832599962523287121810)
tan(18*arccos((tan^2(420/18*pi/180)-213/3)/sqrt(145248)))-sqrt(586632831678141269115189016596917201690588678478429472811665771240231282180853512/11461036947747834700192609320058038125792975816302404862266972458625787589424083192)
tan(18*arccos((tan^2(780/18*pi/180)-213/3)/sqrt(145248)))-sqrt(423747342815321543601214349307116131759616670389350049970330426524985762114599826/11623922436610654425706583987347839195723947824391484285108307803341033109490336878)
tan(18*arccos((tan^2(1140/18*pi/180)-213/3)/sqrt(145248)))-sqrt(20276583614383338613497426856043110661700339402627955479179068355610844228863924/12027393195811592630694300909798912216821864155378206379599459161510408027376072780)
tan(18*arccos((tan^2(2220/18*pi/180)-213/3)/sqrt(145248)))-sqrt(173736907861732222599047031129745923988121837293746990719485492803372813094079259/11873932871564243746708751305525209403495442657487087344359152737062646058510857445)
tan(18*arccos((tan^2(2580/18*pi/180)-213/3)/sqrt(145248)))-sqrt(498291095287844767181951012394652255748562111290265746505243472906027127265938127/11549378684138131202125847324260303071735002383490568588573394756959991744338998577)
tan(18*arccos((tan^2(2940/18*pi/180)-213/3)/sqrt(145248)))-sqrt(610906727412912537543034885809158106813051141084696303959897763690074287281652530/11436763052013063431764763450845797220670513353696138031118740466175944584323284174)
tan(18*arccos((tan^2(3300/18*pi/180)-213/3)/sqrt(145248)))-sqrt(633007263962644773374830690915797168219519435947288814903078397266056348317814894/11414662515463331195932967645739158159264045058833545520175559832599962523287121810)
tan(18*arccos((tan^2(3660/18*pi/180)-213/3)/sqrt(145248)))-sqrt(586632831678141269115189016596917201690588678478429472811665771240231282180853512/11461036947747834700192609320058038125792975816302404862266972458625787589424083192)
tan(18*arccos((tan^2(4020/18*pi/180)-213/3)/sqrt(145248)))-sqrt(423747342815321543601214349307116131759616670389350049970330426524985762114599826/11623922436610654425706583987347839195723947824391484285108307803341033109490336878)
tan(18*arccos((tan^2(4380/18*pi/180)-213/3)/sqrt(145248)))-sqrt(20276583614383338613497426856043110661700339402627955479179068355610844228863924/12027393195811592630694300909798912216821864155378206379599459161510408027376072780)
tan(18*arccos((tan^2(5460/18*pi/180)-213/3)/sqrt(145248)))-sqrt(173736907861732222599047031129745923988121837293746990719485492803372813094079259/11873932871564243746708751305525209403495442657487087344359152737062646058510857445)
tan(18*arccos((tan^2(5820/18*pi/180)-213/3)/sqrt(145248)))-sqrt(498291095287844767181951012394652255748562111290265746505243472906027127265938127/11549378684138131202125847324260303071735002383490568588573394756959991744338998577)
tan(18*arccos((tan^2(6180/18*pi/180)-213/3)/sqrt(145248)))-sqrt(610906727412912537543034885809158106813051141084696303959897763690074287281652530/11436763052013063431764763450845797220670513353696138031118740466175944584323284174)
tan(19*arccos((tan^2(5430/19*pi/180)-73/3)/sqrt(10944)))-sqrt(60992274029829235936789881770896684271856728111091185624912317465/140940014817442931109401473684867479389375980825628229628873061351)
tan(19*arccos((tan^2(5445/19*pi/180)-111/3)/sqrt(33744)))-sqrt(313083658834332915972231406141496781560542795390883479576709411702297974613/81914298982326276333909053156070782810871116690385101001642506182906489003)
tan(19*arccos((tan^2(1500/19*pi/180)-225/3)/sqrt(171456)))-sqrt(3976173729751702974271537711302424472885706900629330676979047661883462392055157189155760/6251131771639420546509676812702505187401519606659928519948717026108098963361187045663824)
tan(19*arccos((tan^2(4740/19*pi/180)-225/3)/sqrt(171456)))-sqrt(10225473927825628198494240685357795491013885015289699658678996379157490374138146808011572/1831573565495322286973838647134169273341491999559538248768308834070981278197426808012)
tan(19*arccos((tan^2(5460/19*pi/180)-225/3)/sqrt(171456)))-sqrt(9977443676893120635751008928144639816539421972457249236754240353058154417304740203408562/249861824498002885030205595860289843747804534832009960173524334933406938111604031411022)
tan(20*arccos((tan^2(45/20*pi/180)-117/3)/sqrt(39520)))-sqrt(41007941230146878322543384365860259058028617420688074101522426275590911313099748/37540738873957077407553945181720355929362883191856995714695173724409088686900252)
tan(20*arccos((tan^2(405/20*pi/180)-117/3)/sqrt(39520)))-sqrt(39924145559494139555651264034505157215336445577402515072972162801851951861095007/38624534544609816174446065513075457772055055035142554743245437198148048138904993)
tan(20*arccos((tan^2(765/20*pi/180)-117/3)/sqrt(39520)))-sqrt(36017795919946954097380346087432292792978945472478832001105865128399838342338550/42530884184157001632716983460148322194412555140066237815111734871600161657661450)
tan(20*arccos((tan^2(1125/20*pi/180)-117/3)/sqrt(39520)))-sqrt(23452969814036555899215951851724987639961543577464647649859682282815976836835790/55095710290067399830881377695855627347429957035080422166357917717184023163164210)
tan(20*arccos((tan^2(2205/20*pi/180)-117/3)/sqrt(39520)))-sqrt(249921894147875252591826505983167077904626623261134193060165457191046840147635/78298758209956080477505503041597447909486873989283935623157434542808953159852365)
tan(20*arccos((tan^2(2565/20*pi/180)-117/3)/sqrt(39520)))-sqrt(28215499937871024947438867092972291576419985427668973280386570492123791927755148/50333180166232930782658462454608323410971515184876096535831029507876208072244852)
tan(20*arccos((tan^2(2925/20*pi/180)-117/3)/sqrt(39520)))-sqrt(37424394616759378085047789749916135934604643841199296532945431455183084378171087/41124285487344577645049539797664479052786856771345773283272168544816915621828913)
tan(20*arccos((tan^2(3285/20*pi/180)-117/3)/sqrt(39520)))-sqrt(40379806690674172540083523045233939252006644042054733093510106712044097112050009/38168873413429783190013806502346675735384856570490336722707493287955902887949991)
tan(20*arccos((tan^2(3645/20*pi/180)-117/3)/sqrt(39520)))-sqrt(41007941230146878322543384365860259058028617420688074101522426275590911313099748/37540738873957077407553945181720355929362883191856995714695173724409088686900252)
tan(20*arccos((tan^2(4005/20*pi/180)-117/3)/sqrt(39520)))-sqrt(39924145559494139555651264034505157215336445577402515072972162801851951861095007/38624534544609816174446065513075457772055055035142554743245437198148048138904993)
tan(20*arccos((tan^2(4365/20*pi/180)-117/3)/sqrt(39520)))-sqrt(36017795919946954097380346087432292792978945472478832001105865128399838342338550/42530884184157001632716983460148322194412555140066237815111734871600161657661450)
tan(20*arccos((tan^2(4725/20*pi/180)-117/3)/sqrt(39520)))-sqrt(23452969814036555899215951851724987639961543577464647649859682282815976836835790/55095710290067399830881377695855627347429957035080422166357917717184023163164210)
tan(20*arccos((tan^2(5805/20*pi/180)-117/3)/sqrt(39520)))-sqrt(249921894147875252591826505983167077904626623261134193060165457191046840147635/78298758209956080477505503041597447909486873989283935623157434542808953159852365)
tan(20*arccos((tan^2(6165/20*pi/180)-117/3)/sqrt(39520)))-sqrt(28215499937871024947438867092972291576419985427668973280386570492123791927755148/50333180166232930782658462454608323410971515184876096535831029507876208072244852)
tan(20*arccos((tan^2(6525/20*pi/180)-117/3)/sqrt(39520)))-sqrt(37424394616759378085047789749916135934604643841199296532945431455183084378171087/41124285487344577645049539797664479052786856771345773283272168544816915621828913)
tan(20*arccos((tan^2(6885/20*pi/180)-117/3)/sqrt(39520)))-sqrt(40379806690674172540083523045233939252006644042054733093510106712044097112050009/38168873413429783190013806502346675735384856570490336722707493287955902887949991)
tan(20*arccos((tan^2(60/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1572235976918537729374136684237065785046113922763710887437724258099486357550066060272733935021/8593774582156218111454718041350166756241067339382646101973304944609344260049933939727266064979)
tan(20*arccos((tan^2(420/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1524318697115985109694637985576033685367823346262749238955370451252841946879205635790130811339/8641691861958770731134216740011198855919357915883607750455658751455988670720794364209869188661)
tan(20*arccos((tan^2(780/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1360788925372025627761006993924295479035716811353840726706862618654986394303866108996759317217/8805221633702730213067847731662937062251464450792516262704166584053844223296133891003240682783)
tan(20*arccos((tan^2(1140/20*pi/180)-237/3)/sqrt(200640)))-sqrt(869508094729577033907975355828293594096670069833930264337141886024598639880908823372487331181/9296502464345178806920879369758938947190511192312426725073887316684231977719091176627512668819)
tan(20*arccos((tan^2(2220/20*pi/180)-237/3)/sqrt(200640)))-sqrt(95101688009690169392294572662862691390478853500614571585210308563815250500443467387316816444/10070908871065065671436560152924369849896702408645742417825818894145015367099556532612683183556)
tan(20*arccos((tan^2(2580/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1099782477838953672135602854579345545190788164075485179604612535837928834262213012514184235932/9066228081235802168693251871007886996096393098070871809806416666870901783337786987485815764068)
tan(20*arccos((tan^2(2940/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1435096403413385121313028694827092882315062901305284347470580087848969033423152363758227845857/8730914155661370719515826030760139658972118360841072641940449114859861584176847636241772154143)
tan(20*arccos((tan^2(3300/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1549278817156161533555903206238628961366773725855224501991477968990824863387981274001506531287/8616731741918594307272951519348603579920407536291132487419551233718005754212018725998493468713)
tan(20*arccos((tan^2(3660/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1572235976918537729374136684237065785046113922763710887437724258099486357550066060272733935021/8593774582156218111454718041350166756241067339382646101973304944609344260049933939727266064979)
tan(20*arccos((tan^2(4020/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1524318697115985109694637985576033685367823346262749238955370451252841946879205635790130811339/8641691861958770731134216740011198855919357915883607750455658751455988670720794364209869188661)
tan(20*arccos((tan^2(4380/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1360788925372025627761006993924295479035716811353840726706862618654986394303866108996759317217/8805221633702730213067847731662937062251464450792516262704166584053844223296133891003240682783)
tan(20*arccos((tan^2(4740/20*pi/180)-237/3)/sqrt(200640)))-sqrt(869508094729577033907975355828293594096670069833930264337141886024598639880908823372487331181/9296502464345178806920879369758938947190511192312426725073887316684231977719091176627512668819)
tan(20*arccos((tan^2(5820/20*pi/180)-237/3)/sqrt(200640)))-sqrt(95101688009690169392294572662862691390478853500614571585210308563815250500443467387316816444/10070908871065065671436560152924369849896702408645742417825818894145015367099556532612683183556)
tan(20*arccos((tan^2(6180/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1099782477838953672135602854579345545190788164075485179604612535837928834262213012514184235932/9066228081235802168693251871007886996096393098070871809806416666870901783337786987485815764068)
tan(20*arccos((tan^2(6540/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1435096403413385121313028694827092882315062901305284347470580087848969033423152363758227845857/8730914155661370719515826030760139658972118360841072641940449114859861584176847636241772154143)
tan(20*arccos((tan^2(6900/20*pi/180)-237/3)/sqrt(200640)))-sqrt(1549278817156161533555903206238628961366773725855224501991477968990824863387981274001506531287/8616731741918594307272951519348603579920407536291132487419551233718005754212018725998493468713)
tan(21*arccos((tan^2(2205/21*pi/180)-123/3)/sqrt(45920)))-sqrt(14254249802356563326949343219018085647098737035661271442581509303458840060172256714172/3891697666817455705435467335635747265454990155547808723334703944541159939827743285828)
tan(21*arccos((tan^2(5805/21*pi/180)-123/3)/sqrt(45920)))-sqrt(12708849319015083493023996959683753607051853131525704792863330470147461128276108393535/5437098150158935539360813594970079305501874059683375373052882777852538871723891606465)
tan(21*arccos((tan^2(2220/21*pi/180)-249/3)/sqrt(232960)))-sqrt(11683636901782635231626833298372313734384086522171098816329678106980110459132815835348042793313122358/55838560530539884474259100887553691198457512538111451904948229836589629777008164651957206686877642)
tan(21*arccos((tan^2(5460/21*pi/180)-249/3)/sqrt(232960)))-sqrt(4244907699916117525770409256010864769993423473711845453253986964838782213008589314533770703315102420/7494567762397057590330683143249002655589120560997364814980639371977917875901234685466229296684897580)
tan(22*arccos((tan^2(750/22*pi/180)-85/3)/sqrt(17248)))-sqrt(91814395447438480455483216217295939143481064889895177697100316846081666753585054/4602305791248458547764050003218209843072054399399649031284479223292707643490)
tan(22*arccos((tan^2(1110/22*pi/180)-85/3)/sqrt(17248)))-sqrt(88880631235240075444363812622065235910170973552616120290191912563443845801723164/2938366517989653469667167645233921443153163391678457055939688761861113659505380)
tan(22*arccos((tan^2(1470/22*pi/180)-85/3)/sqrt(17248)))-sqrt(39086947908935453872587032002779929863710420844615268239331709806300773755373570/52732049844294275041443948264519227489613716099679309106799891519004185705854974)
tan(22*arccos((tan^2(2550/22*pi/180)-85/3)/sqrt(17248)))-sqrt(57921831621876430136623094608985316206130981085895583473334829972877897114086783/33897166131353298777407885658313841147193155858398993872796771352427062347141761)
tan(22*arccos((tan^2(2910/22*pi/180)-85/3)/sqrt(17248)))-sqrt(90133464751461451766086216000983462780872665244041194651157452642160140536661521/1685533001768277147944764266315694572451471700253382694974148683144818924567023)
tan(22*arccos((tan^2(4710/22*pi/180)-85/3)/sqrt(17248)))-sqrt(91814395447438480455483216217295939143481064889895177697100316846081666753585054/4602305791248458547764050003218209843072054399399649031284479223292707643490)
tan(22*arccos((tan^2(5070/22*pi/180)-85/3)/sqrt(17248)))-sqrt(88880631235240075444363812622065235910170973552616120290191912563443845801723164/2938366517989653469667167645233921443153163391678457055939688761861113659505380)
tan(22*arccos((tan^2(5430/22*pi/180)-85/3)/sqrt(17248)))-sqrt(39086947908935453872587032002779929863710420844615268239331709806300773755373570/52732049844294275041443948264519227489613716099679309106799891519004185705854974)
tan(22*arccos((tan^2(6510/22*pi/180)-85/3)/sqrt(17248)))-sqrt(57921831621876430136623094608985316206130981085895583473334829972877897114086783/33897166131353298777407885658313841147193155858398993872796771352427062347141761)
tan(22*arccos((tan^2(6870/22*pi/180)-85/3)/sqrt(17248)))-sqrt(90133464751461451766086216000983462780872665244041194651157452642160140536661521/1685533001768277147944764266315694572451471700253382694974148683144818924567023)
tan(22*arccos((tan^2(45/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3391389331166948910577218790876858604157106255977001715031134497225698686658026311726545249/1443892423649674053938685627281420441571708097847596365506258136699728992969250089714697887)
tan(22*arccos((tan^2(405/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3344021664092801262715990384535383183708906089182537926351958198336216923706488224430758148/1491260090723821701799914033622895862019908264642060154185434435589210755920788177010484988)
tan(22*arccos((tan^2(765/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3180441416707302225334538304093832711500453855367134250468581482238868335587002201510371363/1654840338109320739181366114064446334228360498457463830068811151686559344040274199930871773)
tan(22*arccos((tan^2(1125/22*pi/180)-129/3)/sqrt(52976)))-sqrt(2695840277816294791440682868082565586711540054791224638342296521822506054031131374090608551/2139441477000328173075221550075713459017274299033373442195096112102921625596145027350634585)
tan(22*arccos((tan^2(1485/22*pi/180)-129/3)/sqrt(52976)))-sqrt(829414499123423283749112810755103613530564829385323763757707581724235971772461471360288903/4005867255693199680766791607403175432198249524439274316779685052201191707854814930080954233)
tan(22*arccos((tan^2(2205/22*pi/180)-129/3)/sqrt(52976)))-sqrt(4113982815984268985786905495677893374113902612419162188139425691525517390541125208696179152/721298938832353978728998922480385671614911741405435892397966942399910289086151192745063984)
tan(22*arccos((tan^2(2565/22*pi/180)-129/3)/sqrt(52976)))-sqrt(1564673137664025689316988605695974127357394427304273908495749592045002764225810854557028339/3270608617152597275198915812462304918371419926520324172041643041880424915401465546884214797)
tan(22*arccos((tan^2(2925/22*pi/180)-129/3)/sqrt(52976)))-sqrt(2875195009878961586345983223229238549503924848374729078850836522183587724797508336089692659/1960086744937661378169921194929040496224889505449869001686556111741839954829768065351550477)
tan(22*arccos((tan^2(3285/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3238220732005889590304570117394439476000740344731699450210721125496400931521072485976983146/1597061022810733374211334300763839569728074009092898630326671508429026748106203915464259990)
tan(22*arccos((tan^2(3645/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3363812900119698000250010797194395821873290739919675420163736629080282904666355723537067566/1471468854696924964265893620963883223855523613904922660373656004845144774960920677904175570)
tan(22*arccos((tan^2(4005/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3391389331166948910577218790876858604157106255977001715031134497225698686658026311726545249/1443892423649674053938685627281420441571708097847596365506258136699728992969250089714697887)
tan(22*arccos((tan^2(4365/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3344021664092801262715990384535383183708906089182537926351958198336216923706488224430758148/1491260090723821701799914033622895862019908264642060154185434435589210755920788177010484988)
tan(22*arccos((tan^2(4725/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3180441416707302225334538304093832711500453855367134250468581482238868335587002201510371363/1654840338109320739181366114064446334228360498457463830068811151686559344040274199930871773)
tan(22*arccos((tan^2(5085/22*pi/180)-129/3)/sqrt(52976)))-sqrt(2695840277816294791440682868082565586711540054791224638342296521822506054031131374090608551/2139441477000328173075221550075713459017274299033373442195096112102921625596145027350634585)
tan(22*arccos((tan^2(5445/22*pi/180)-129/3)/sqrt(52976)))-sqrt(829414499123423283749112810755103613530564829385323763757707581724235971772461471360288903/4005867255693199680766791607403175432198249524439274316779685052201191707854814930080954233)
tan(22*arccos((tan^2(6165/22*pi/180)-129/3)/sqrt(52976)))-sqrt(4113982815984268985786905495677893374113902612419162188139425691525517390541125208696179152/721298938832353978728998922480385671614911741405435892397966942399910289086151192745063984)
tan(22*arccos((tan^2(6525/22*pi/180)-129/3)/sqrt(52976)))-sqrt(1564673137664025689316988605695974127357394427304273908495749592045002764225810854557028339/3270608617152597275198915812462304918371419926520324172041643041880424915401465546884214797)
tan(22*arccos((tan^2(6885/22*pi/180)-129/3)/sqrt(52976)))-sqrt(2875195009878961586345983223229238549503924848374729078850836522183587724797508336089692659/1960086744937661378169921194929040496224889505449869001686556111741839954829768065351550477)
tan(22*arccos((tan^2(7245/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3238220732005889590304570117394439476000740344731699450210721125496400931521072485976983146/1597061022810733374211334300763839569728074009092898630326671508429026748106203915464259990)
tan(22*arccos((tan^2(7605/22*pi/180)-129/3)/sqrt(52976)))-sqrt(3363812900119698000250010797194395821873290739919675420163736629080282904666355723537067566/1471468854696924964265893620963883223855523613904922660373656004845144774960920677904175570)
tan(22*arccos((tan^2(60/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4541197980621943934131268448464972877190247758320960530179555745627248370122269895332373525654100385683546/11095988750391904716692505525566337802959147922170635782516506571138935437543714505839136394775487057274790)
tan(22*arccos((tan^2(420/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4469522863384890751494561902341111268988850874721191381972515764215061465541805371359514925085468033818887/11167663867628957899329212071690199411160544805770404930723546552551122342124179029811994995344119409139449)
tan(22*arccos((tan^2(780/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4235797118805450196901208943841175464622847426310928339694850683765947259594591120844713120769078529557189/11401389612208398453922565030190135215526548254180667973001211633000236548071393280326796799660508913401147)
tan(22*arccos((tan^2(1140/22*pi/180)-261/3)/sqrt(268576)))-sqrt(3589100300271511346190764006154478565495314615014700070443329593055508249492770944277961243127479587242315/12048086430742337304633009967876832114654081065476896242252732723710675558173213456893548677302107855716021)
tan(22*arccos((tan^2(1500/22*pi/180)-261/3)/sqrt(268576)))-sqrt(1372831610025052783628096230820889527347596421657125355899544435423891954191543115788634172349251969718372/14264355120988795867195677743210421152801799258834470956796517881342291853474441285382875748080335473239964)
tan(22*arccos((tan^2(2580/22*pi/180)-261/3)/sqrt(268576)))-sqrt(2453806242924883853279670924892375512929266917453799092820289555847495032135624709005077789581839462057293/13183380488088964797544103049138935167220128763037797219875772760918688775530359692166432130847747980901043)
tan(22*arccos((tan^2(2940/22*pi/180)-261/3)/sqrt(268576)))-sqrt(3885892334838703982910732592395692009236847555535104549341656330689576162093564396063309124230466977936524/11751294396175144667913041381635618670912548124956491763354405986076607645572420005108200796199120465021812)
tan(22*arccos((tan^2(3300/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4340170328107318093590173749243282639181468092236473668458844873955593504706327335310320489921553939943867/11297016402906530557233600224788028040967927588255122644237217442810590302959657065861189430508033503014469)
tan(22*arccos((tan^2(3660/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4506644177739295503107715571655092391188893457068290168465164171138655375770656664474948511791264318808168/11130542553274553147716058402376218288960502223423306144230898145627528431895327736696561408638323124150168)
tan(22*arccos((tan^2(4020/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4541197980621943934131268448464972877190247758320960530179555745627248370122269895332373525654100385683546/11095988750391904716692505525566337802959147922170635782516506571138935437543714505839136394775487057274790)
tan(22*arccos((tan^2(4380/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4469522863384890751494561902341111268988850874721191381972515764215061465541805371359514925085468033818887/11167663867628957899329212071690199411160544805770404930723546552551122342124179029811994995344119409139449)
tan(22*arccos((tan^2(4740/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4235797118805450196901208943841175464622847426310928339694850683765947259594591120844713120769078529557189/11401389612208398453922565030190135215526548254180667973001211633000236548071393280326796799660508913401147)
tan(22*arccos((tan^2(5100/22*pi/180)-261/3)/sqrt(268576)))-sqrt(3589100300271511346190764006154478565495314615014700070443329593055508249492770944277961243127479587242315/12048086430742337304633009967876832114654081065476896242252732723710675558173213456893548677302107855716021)
tan(22*arccos((tan^2(5460/22*pi/180)-261/3)/sqrt(268576)))-sqrt(1372831610025052783628096230820889527347596421657125355899544435423891954191543115788634172349251969718372/14264355120988795867195677743210421152801799258834470956796517881342291853474441285382875748080335473239964)
tan(22*arccos((tan^2(6540/22*pi/180)-261/3)/sqrt(268576)))-sqrt(2453806242924883853279670924892375512929266917453799092820289555847495032135624709005077789581839462057293/13183380488088964797544103049138935167220128763037797219875772760918688775530359692166432130847747980901043)
tan(22*arccos((tan^2(6900/22*pi/180)-261/3)/sqrt(268576)))-sqrt(3885892334838703982910732592395692009236847555535104549341656330689576162093564396063309124230466977936524/11751294396175144667913041381635618670912548124956491763354405986076607645572420005108200796199120465021812)
tan(22*arccos((tan^2(7260/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4340170328107318093590173749243282639181468092236473668458844873955593504706327335310320489921553939943867/11297016402906530557233600224788028040967927588255122644237217442810590302959657065861189430508033503014469)
tan(22*arccos((tan^2(7620/22*pi/180)-261/3)/sqrt(268576)))-sqrt(4506644177739295503107715571655092391188893457068290168465164171138655375770656664474948511791264318808168/11130542553274553147716058402376218288960502223423306144230898145627528431895327736696561408638323124150168)
tan(23*arccos((tan^2(2205/23*pi/180)-135/3)/sqrt(60720)))-sqrt(3533945993273888671611565262775765339855372698324887982086519813507985532584148755282697241080/1473024852672087745171904191520086981326179849123431692131903017335662497667415851244717302758920)
tan(23*arccos((tan^2(6525/23*pi/180)-135/3)/sqrt(60720)))-sqrt(1128046557419422190285915580408993052327444847031052042325662537374342854121488678960329159238373/348512241245939443557600176373869694338590374790704537788326999774827629078511321039670840761627)
tan(23*arccos((tan^2(1860/23*pi/180)-273/3)/sqrt(307648)))-sqrt(7623144484321008405151977222572496219566002177140219973065329853700102710835454946448499485137575557256844452155/16247514584697426778389172138900294989543376108769771935136902646034479151142726736194484164294883437448211160773)

Vinicius
Nov/15th/2,020
Vinicius Claudino Ferraz
2020-11-17 01:00:42 UTC
Today's release is worried with plus or minus sign.

cos(7*arccos((tan^2(30/7*pi/180)-25/3)/sqrt(448))))-sqrt(20693880329219/112^7)
cos(7*arccos((tan^2(390/7*pi/180)-25/3)/sqrt(448))))-sqrt(169516599755971/112^7)
cos(7*arccos((tan^2(750/7*pi/180)-25/3)/sqrt(448))))+sqrt(96371132936875/112^7)
cos(7*arccos((tan^2(1110/7*pi/180)-25/3)/sqrt(448))))-sqrt(28055831369342/112^7)
cos(7*arccos((tan^2(210*pi/180)-25/3)/sqrt(448))))-377/7^3.5
cos(7*arccos((tan^2(2190/7*pi/180)-25/3)/sqrt(448))))-sqrt(96520791404839/112^7)
cos(7*arccos((tan^2(45/7*pi/180)-13)/sqrt(1456))))-sqrt(360150403588837119/364^7)
cos(7*arccos((tan^2(405/7*pi/180)-13)/sqrt(1456))))-sqrt(733458804863979016/364^7)
cos(7*arccos((tan^2(765/7*pi/180)-13)/sqrt(1456))))-sqrt(510386226351227899/364^7)
cos(7*arccos((tan^2(1125/7*pi/180)-13)/sqrt(1456))))-sqrt(378117845233971187/364^7)
cos(7*arccos((tan^2(1485/7*pi/180)-13)/sqrt(1456))))-sqrt(422912491224802069/364^7)
cos(7*arccos((tan^2(315*pi/180)-13)/sqrt(1456))))-5640807/91^3.5
cos(7*arccos((tan^2(60/7*pi/180)-26)/sqrt(7616))))-sqrt(59303434810012969818158/1904^7)
cos(7*arccos((tan^2(60*pi/180)-26)/sqrt(7616))))-17075643/119^3.5
cos(7*arccos((tan^2(780/7*pi/180)-26)/sqrt(7616))))-sqrt(89984964054295008542780/1904^7)
cos(7*arccos((tan^2(1140/7*pi/180)-26)/sqrt(7616))))-sqrt(59829603191239033829971/1904^7)
cos(7*arccos((tan^2(1500/7*pi/180)-26)/sqrt(7616))))-sqrt(62477697780247763172928/1904^7)
cos(7*arccos((tan^2(2220/7*pi/180)-26)/sqrt(7616))))-sqrt(65236980476831432091318/1904^7)
cos(7*arccos((tan^2(120/7*pi/180)-27)/sqrt(7616))))-sqrt(59829603191239033829971/1904^7)
cos(7*arccos((tan^2(480/7*pi/180)-27)/sqrt(7616))))-sqrt(89984964054295008542780/1904^7)
cos(7*arccos((tan^2(120*pi/180)-27)/sqrt(7616))))-17075643/119^3.5
cos(7*arccos((tan^2(1200/7*pi/180)-27)/sqrt(7616))))-sqrt(59303434810012969818158/1904^7)
cos(7*arccos((tan^2(1560/7*pi/180)-27)/sqrt(7616))))-sqrt(65236980476831432091318/1904^7)
cos(7*arccos((tan^2(2280/7*pi/180)-27)/sqrt(7616))))-sqrt(62477697780247763172928/1904^7)
cos(7*arccos((tan^2(135/7*pi/180)-13)/sqrt(1456))))-sqrt(378117845233971187/364^7)
cos(7*arccos((tan^2(495/7*pi/180)-13)/sqrt(1456))))-sqrt(510386226351227899/364^7)
cos(7*arccos((tan^2(855/7*pi/180)-13)/sqrt(1456))))-sqrt(733458804863979016/364^7)
cos(7*arccos((tan^2(1215/7*pi/180)-13)/sqrt(1456))))-sqrt(360150403588837119/364^7)
cos(7*arccos((tan^2(225*pi/180)-13)/sqrt(1456))))-5640807/91^3.5
cos(7*arccos((tan^2(2295/7*pi/180)-13)/sqrt(1456))))-sqrt(422912491224802069/364^7)
cos(7*arccos((tan^2(150/7*pi/180)-25/3)/sqrt(448))))-sqrt(28055831369342/112^7)
cos(7*arccos((tan^2(510/7*pi/180)-25/3)/sqrt(448))))+sqrt(96371132936875/112^7)
cos(7*arccos((tan^2(870/7*pi/180)-25/3)/sqrt(448))))-sqrt(169516599755971/112^7)
cos(7*arccos((tan^2(1230/7*pi/180)-25/3)/sqrt(448))))-sqrt(20693880329219/112^7)
cos(7*arccos((tan^2(1590/7*pi/180)-25/3)/sqrt(448))))-sqrt(96520791404839/112^7)
cos(7*arccos((tan^2(330*pi/180)-25/3)/sqrt(448))))-377/7^3.5
cos(7*arccos((tan^2(180/7*pi/180)-6)/sqrt(140))))+sqrt(10837590372/35^7)
cos(7*arccos((tan^2(900/7*pi/180)-6)/sqrt(140))))-sqrt(12520372122/35^7)
cos(7*arccos((tan^2(180*pi/180)-6)/sqrt(140))))+139233/35^3.5
cos(7*arccos((tan^2(1620/7*pi/180)-6)/sqrt(140))))-sqrt(12520372122/35^7)
cos(7*arccos((tan^2(2340/7*pi/180)-6)/sqrt(140))))+sqrt(10837590372/35^7)
cos(7*arccos((tan^2(30*pi/180)-25/3)/sqrt(448))))-377/7^3.5
cos(7*arccos((tan^2(930/7*pi/180)-25/3)/sqrt(448))))-sqrt(96520791404839/112^7)
cos(7*arccos((tan^2(1290/7*pi/180)-25/3)/sqrt(448))))-sqrt(20693880329219/112^7)
cos(7*arccos((tan^2(1650/7*pi/180)-25/3)/sqrt(448))))-sqrt(169516599755971/112^7)
cos(7*arccos((tan^2(2010/7*pi/180)-25/3)/sqrt(448))))+sqrt(96371132936875/112^7)
cos(7*arccos((tan^2(2370/7*pi/180)-25/3)/sqrt(448))))-sqrt(28055831369342/112^7)
cos(7*arccos((tan^2(225/7*pi/180)-13)/sqrt(1456))))-sqrt(422912491224802069/364^7)
cos(7*arccos((tan^2(135*pi/180)-13)/sqrt(1456))))-5640807/91^3.5
cos(7*arccos((tan^2(1305/7*pi/180)-13)/sqrt(1456))))-sqrt(360150403588837119/364^7)
cos(7*arccos((tan^2(1665/7*pi/180)-13)/sqrt(1456))))-sqrt(733458804863979016/364^7)
cos(7*arccos((tan^2(2025/7*pi/180)-13)/sqrt(1456))))-sqrt(510386226351227899/364^7)
cos(7*arccos((tan^2(2385/7*pi/180)-13)/sqrt(1456))))-sqrt(378117845233971187/364^7)
cos(7*arccos((tan^2(240/7*pi/180)-26)/sqrt(7616))))-sqrt(62477697780247763172928/1904^7)
cos(7*arccos((tan^2(960/7*pi/180)-26)/sqrt(7616))))-sqrt(65236980476831432091318/1904^7)
cos(7*arccos((tan^2(1320/7*pi/180)-26)/sqrt(7616))))-sqrt(59303434810012969818158/1904^7)
cos(7*arccos((tan^2(240*pi/180)-26)/sqrt(7616))))-17075643/119^3.5
cos(7*arccos((tan^2(2040/7*pi/180)-26)/sqrt(7616))))-sqrt(89984964054295008542780/1904^7)
cos(7*arccos((tan^2(2400/7*pi/180)-26)/sqrt(7616))))-sqrt(59829603191239033829971/1904^7)
cos(7*arccos((tan^2(300/7*pi/180)-27)/sqrt(7616))))-sqrt(65236980476831432091318/1904^7)
cos(7*arccos((tan^2(1020/7*pi/180)-27)/sqrt(7616))))-sqrt(62477697780247763172928/1904^7)
cos(7*arccos((tan^2(1380/7*pi/180)-27)/sqrt(7616))))-sqrt(59829603191239033829971/1904^7)
cos(7*arccos((tan^2(1740/7*pi/180)-27)/sqrt(7616))))-sqrt(89984964054295008542780/1904^7)
cos(7*arccos((tan^2(300*pi/180)-27)/sqrt(7616))))-17075643/119^3.5
cos(7*arccos((tan^2(2460/7*pi/180)-27)/sqrt(7616))))-sqrt(59303434810012969818158/1904^7)
cos(7*arccos((tan^2(45*pi/180)-12)/sqrt(1456))))-5640807/91^3.5
cos(7*arccos((tan^2(1035/7*pi/180)-12)/sqrt(1456))))-sqrt(422912491224802069/364^7)
cos(7*arccos((tan^2(1395/7*pi/180)-12)/sqrt(1456))))-sqrt(378117845233971187/364^7)
cos(7*arccos((tan^2(1755/7*pi/180)-12)/sqrt(1456))))-sqrt(510386226351227899/364^7)
cos(7*arccos((tan^2(2115/7*pi/180)-12)/sqrt(1456))))-sqrt(733458804863979016/364^7)
cos(7*arccos((tan^2(2475/7*pi/180)-12)/sqrt(1456))))-sqrt(360150403588837119/364^7)
cos(7*arccos((tan^2(330/7*pi/180)-25/3)/sqrt(448))))-sqrt(96520791404839/112^7)
cos(7*arccos((tan^2(150*pi/180)-25/3)/sqrt(448))))-377/7^3.5
cos(7*arccos((tan^2(1410/7*pi/180)-25/3)/sqrt(448))))-sqrt(28055831369342/112^7)
cos(7*arccos((tan^2(1770/7*pi/180)-25/3)/sqrt(448))))+sqrt(96371132936875/112^7)
cos(7*arccos((tan^2(2130/7*pi/180)-25/3)/sqrt(448))))-sqrt(169516599755971/112^7)
cos(7*arccos((tan^2(2490/7*pi/180)-25/3)/sqrt(448))))-sqrt(20693880329219/112^7)
cos(7*arccos((tan^2(360/7*pi/180)-6)/sqrt(140))))-sqrt(12520372122/35^7)
cos(7*arccos((tan^2(1080/7*pi/180)-6)/sqrt(140))))+sqrt(10837590372/35^7)
cos(7*arccos((tan^2(1440/7*pi/180)-6)/sqrt(140))))+sqrt(10837590372/35^7)
cos(7*arccos((tan^2(2160/7*pi/180)-6)/sqrt(140))))-sqrt(12520372122/35^7)
cos(7*arccos((tan^2(360*pi/180)-6)/sqrt(140))))+139233/35^3.5

Vinicius
Nov/16th/2,020
mitchr...@gmail.com
2020-11-17 01:55:29 UTC
Attach a minus sign.
Vinicius Claudino Ferraz
2020-11-20 18:58:26 UTC
Post by ***@gmail.com
Attach a minus sign.
Such a comment doesn't help. Well, I tried to help, rather than doing crosswords.

This maledictus group doesn't support attaching. That's good, or else John Gabriel would use the entire disk in terabytes.

The plus or minus sign... for the sines and cosssines... only define the argument of a complex number, in the interval J = [0, 2 pi).

1) there is at wikipedia solution to some quintics with hypergeometric 2 F_1( ... )

2) I tried to do this with a single sextic and failed. Does somebody know?

Maybe 300 constants may be found by a supercomputer.

Att

Vinicius
Nov/20th/2,020 anno Domini
Vinicius Claudino Ferraz
2020-11-20 21:11:45 UTC
Hello, intuitionists. Where are you now?

The fourth line of my program's output says that:

cos(3*arccos((tan^2(20*pi/180)-10)/sqrt(448)))-1184/112^1.5 = 0

There were some people here like bkk saying that 20 degrees does not exist.

What's up?
Bassam Karzeddin
2020-11-22 12:48:50 UTC
Post by Vinicius Claudino Ferraz
Hello, intuitionists. Where are you now?
cos(3*arccos((tan^2(20*pi/180)-10)/sqrt(448)))-1184/112^1.5 = 0
There were some people here like bkk saying that 20 degrees does not exist.
What's up?
Yes, that is absolutely true, the angle (pi/9 = 20 Degrees) doesn't **STRICTLY** exist (FOR 100% SURE)

That was only and **HISTORICALLY** claimed by me known as BKK or in short (Bassam Karzeddin), with several elementary public published proofs in my public ***STOLEN*** profiles by proffessional academic mathematicians ***THIEVES*** under the sunlight in this open era of supercomputer era and immediate **GLOBAL** communications where I can't access them on sci. math any more...

However, if the angle (pi/9) does indeed exist, then certainly Wenzel (in 1826) and many others before and after him would have easily and ***EXACTLY*** constructed it, but it doesn't

What the true traitors of ALL human true knowledge usually do is only ***CHATTING*** the so innocent minds since their early childhood in schools by their ***ENDLESS*** brain fart **LOGIC, PHILOSOPHY, PHYSICS and MATHEMATICS*** as well by exploiting the endless *DENSITY* of real existing (constructible) numers or equivalently the density of existing (constructible) angles and make the **FOOLS** believe blindly in their very fake talents (that are good enough for earthy little practical carpentry works)

Where this type of **CHEAT** is being run smoothly in the people heads for thousands of years nowadays

Can you kindly provide me with a valid link to my own public published profile (on sci. math) if not (completely stolen and deleted yet)?

Thanks for anyone doing this favour for me

Bassam Karzeddin
Vinicius Claudino Ferraz
2020-11-22 19:31:52 UTC
For BKK and for all the intuitionists:

That shows you are not conscious.

https://www.wolframalpha.com/input/?i=cos%283*arccos%28%28tan%5E2%2840*pi%2F180%29-11%29%2Fsqrt%28448%29%29%29-1184%2F112%5E1.5

The result was not zero, but is zero now. Let me isolate for you.

cos(3*arccos((tan^2(40*pi/180)-11)/sqrt(448)))-1184/112^1.5 = 0

cos(3*arccos((tan^2(40*pi/180)-11)/sqrt(448)))=1184/112^1.5

3*arccos((tan^2(40*pi/180)-11)/sqrt(448))=2*pi-arccos(1184/112^1.5)

arccos((tan^2(40*pi/180)-11)/sqrt(448))=2/3*pi-1/3*arccos(1184/112^1.5)

(tan^2(40*pi/180)-11)/sqrt(448)=cos(2/3*pi-1/3*arccos(1184/112^1.5))

tan^2(40*pi/180)-11=sqrt(448)*cos(2/3*pi-1/3*arccos(1184/112^1.5))

tan^2(40*pi/180)=11+sqrt(448)*cos(2/3*pi-1/3*arccos(1184/112^1.5))

https://www.wolframalpha.com/input/?i=tan%5E2%2840*pi%2F180%29%3D11%2Bsqrt%28448%29*cos%282%2F3*pi-1%2F3*arccos%281184%2F112%5E1.5%29%29

Wolfram says: True. Is it good for you now, intuitionists? It was false yesterday.

I know, you are going to say that the dual angles do not exist either.

Sincerely,

Vinicius
Vinicius Claudino Ferraz
2020-11-22 20:30:27 UTC
Once again. LOL. My program rounded 11 to 10. This number is a twin of the other.

https://www.wolframalpha.com/input/?i=tan%5E2%2820*pi%2F180%29%3D11%2Bsqrt%28448%29*cos%282%2F3*pi%2B1%2F3*arccos%281184%2F112%5E1.5%29%29
Post by Vinicius Claudino Ferraz
Wolfram says: True. Is it good for you now, intuitionists? It was false yesterday.
I know, you are going to say that the dual angles do not exist either.
Sincerely,
Vinicius
Vinicius Claudino Ferraz
2020-11-22 21:41:17 UTC
I searched here for 1 degree and 2 degrees, they are in my hands LOL. But Python's precision beats wolfram alpha's one.

https://www.wolframalpha.com/input/?i=cos%28+15*arccos%28%28tan%5E2%284*pi%2F180%29-59%29%2Fsqrt%2882880%29%29%29-sqrt%28116825359739263661544053232446390794175853636352211324018330370%2F20720%5E15%29

4 degrees are cute, aren't they?
Post by Vinicius Claudino Ferraz
Sincerely,
Vinicius
Vinicius Claudino Ferraz
2020-11-23 16:12:27 UTC
#censored

Since 1 degree has been found, I'm searching for 1 rad, and the angles 1/pi, sqrt 2, sqrt 3, sqrt pi, 1/sqrt(pi), pi^2...

Post by Vinicius Claudino Ferraz
Sincerely,
Vinicius
Vinicius Claudino Ferraz
2020-11-24 21:31:38 UTC
People o' the wowd,

If 1 degree is unconstructible, I have found infinitely many dual numbers for it.
The things remain unconstructible, of course, but they are gathered. It is like a table many-to-many.

I'm using my own factory. It doesn't matter if that formula can be found in a obscure papyrus.
I'm doing charity here. It's my formula. Yesterday I divided 1 degree into N parts.
Today I divided 180 degrees by pi.
Tomorrow I will divide by a + bi = 2x2_ (a, b, -b, a) and matrixes n x n.

I know the discriminant for p = 2 and 3 and 4. But I don't know for any natural p. Nor any real p. Nor any complex p. Neither any complex p = a + bi.

Exercise for statisctic analysis. Which numbers are these?
Mainly tan Arg z = ??? is it an algebraic fraction p(x)/q(x) of the parameter x = tan(m)?
My theorem holds sometimes. We want to know when.
What is the Historical Philosopher who said the Truth is Essential, "sometimes"?

Stomachal joke.
Definition:
Division of the number a by the matrix B.
Easy. Multiply a by B⁻¹.
If the divisor isn't invertible, then raise exception DIVISION_BY_SINGULAR.

Sincerely,

Vinicius
Bassam Karzeddin
2020-11-25 14:26:13 UTC
Post by Vinicius Claudino Ferraz
People o' the wowd,
If 1 degree is unconstructible,
Of course, it is impossible to construct the non-existing angles like one degree
Post by Vinicius Claudino Ferraz
I have found infinitely many dual numbers for it.
You do precisely mean that you have found many approximated angles but never that angle in your mind around it that are existing (constructible) angles, where the so innocent doesn't recognize the fact from the illusion
However, this wasn't your own sin but so many many others before and after you as well since ALL of them refuse *stubbornly* to understand the fact of their own universal idiocy and like to be recognized as discoverers by exploiting very badly the average intelligence of the very foolish mainstreams especially among the academic proffessional mathematician's experts **STRICTLY**

This is in-fact the main talent of a skilled carpenter that is available in plenty and much before BC
Post by Vinicius Claudino Ferraz
The things remain unconstructible, of course, but they are gathered. It is like a table many-to-many.
The main thing is that you are mentally disabled to understand the fact despite pronouncing it earlier
Post by Vinicius Claudino Ferraz
I'm using my own factory. It doesn't matter if that formula can be found in a obscure papyrus.
Don't use anything or any tools on a very elementary settled and published issues about the non-existing objects in mathematics
Post by Vinicius Claudino Ferraz
I'm doing charity here. It's my formula. Yesterday I divided 1 degree into N parts.
That is a very interesting Joke indeed
Post by Vinicius Claudino Ferraz
Today I divided 180 degrees by pi.
Don't waste your life after SO many illusions in mathematics and follow the facts stated *strictly* by me
Post by Vinicius Claudino Ferraz
Tomorrow I will divide by a + bi = 2x2_ (a, b, -b, a) and matrixes n x n.
You would certainly go mad after the many illusions in mathematics had been well-explained and rigorously proven even to mid-school students and interested laypersons and amateurs as well
Post by Vinicius Claudino Ferraz
I know the discriminant for p = 2 and 3 and 4. But I don't know for any natural p. Nor any real p. Nor any complex p. Neither any complex p = a + bi.
You don't know anything it seems
Post by Vinicius Claudino Ferraz
Exercise for statisctic analysis. Which numbers are these?
Mainly tan Arg z = ??? is it an algebraic fraction p(x)/q(x) of the parameter x = tan(m)?
My theorem holds sometimes. We want to know when.
What is the Historical Philosopher who said the Truth is Essential, "sometimes"?
Relax, there is no historical or living philosopher, logicians, etc
Post by Vinicius Claudino Ferraz
Stomachal joke.
Division of the number a by the matrix B.
Easy. Multiply a by B⁻¹.
If the divisor isn't invertible, then raise exception DIVISION_BY_SINGULAR.
All that you have learnt is mostly better slightly than a rubbish
Post by Vinicius Claudino Ferraz
Sincerely,
You are never sincere ...
Post by Vinicius Claudino Ferraz
Vinicius
So what? Like many LITTLE dwarfs before and after you too .... (FOR SURE)

Bassam Karzeddin
Vinicius Claudino Ferraz
2020-11-25 21:04:28 UTC
Post by Bassam Karzeddin
You do precisely mean that you have found many approximated angles
it's exact. the precision is for all epsilon > 0.
Post by Bassam Karzeddin
Post by Vinicius Claudino Ferraz
I'm doing charity here. It's my formula. Yesterday I divided 1 degree into N parts.
That is a very interesting Joke indeed
It's not a joke. Try me. What do you want to divide by what?
I have brothers for int N degrees divided by int P.
I have brothers for real R radians divided by real S.
It's not necessary continued fractions. It's exact.
Post by Bassam Karzeddin
Relax, there is no historical or living philosopher, logicians, etc
I'm trying to get into the History Channel.
Post by Bassam Karzeddin
You are never sincere ...
Always.
Post by Bassam Karzeddin
So what? Like many LITTLE dwarfs before and after you too
Better than crosswords, I said. Last month I became little bachelor. LOL.
Post by Bassam Karzeddin
Bassam Karzeddin
It begins funny and it ends deep.

First 45 degrees whose tangent is equal to 1 is brother of arctan 2/11. (modulo 3)
I understand that 45/3 is drawable with ruler and compass exactly like (11 + 2 i)^(1/3) too.

After that I divided the angles into 4 parts:
1) 30 degrees, 60 degrees whose tangent is square root of a fraction. I found their brothers are square roots too.
2) 36 degrees, whose tangent is 1 + sqrt(5) over 2. Its brother is ugly.
3) 20 degrees whose brother is a square root. I understand that it is not drawable with ruler and compass exactly like its brother.
4) the contrary case.

The formula is public.
The code is public.
The system printed 1 rad as a function of tan 180 degrees = 0 because

#######################
# (Mod z)^2 = (b^2/4)^pi
# Is it useful?
#######################

delta = 57.2957795130823208767981548141051703324054724665643215491602438612
m := delta*pi/180 = 1.0
a = 2.14159265358979323846264338327950288419716939937510582097494459231
b = 2.28819193772973334730334233011235469736404949272448242491655788537
Re z = -0.255850207340132755840527680342065596497993254781093526425435728723
Im z = -1.50479366472820335907683783342246038389816099748423934574228540542
Mod z = 1.52638897467261778545080069225212739008757198295924299955753400954
tan Arg z = 5.88154170509503777861655781554197112776618217919714472669079108958
cc = -0.16761796081173076806932060316305210351250435988623680697048790365 = [fraction] beta/gamma^p sometimes
==> cos( 3.14159265358979323846264338327950288419716939937510582097494459231 * arccos((tan^2(m)-a)/b)) = cc

tan Arg z is, by definition, the brother of 180 degrees (modulo pi).
Vinicius Claudino Ferraz
2020-11-25 21:12:28 UTC
tan x ~ tan y (mod p), x, y, p are real

I want something for testing division by seven. Simple. 45 degrees *7. Divide 315 by 7.

tan (45° * p) ~ tan (sqrt b/a) (mod p), p, a, b are int
mitchr...@gmail.com
2020-11-26 01:14:34 UTC
A zero angle in a trig table doesn't belong.
No polygon can have a zero angle...
They need to make a correction
and give an infinitesimal angle
Vinicius Claudino Ferraz
2020-11-26 17:28:16 UTC
oh my god x = 0 = 2 pi = 4 pi = 360° = 720° are working.

0.00000: cos( 3*arccos((tan^2(0*pi/180)-2)/2))+1
0.00000: cos( 5*arccos((tan^2(0*pi/180)-4)/sqrt(40)))-sqrt(242/5^5)
0.00000: cos( 7*arccos((tan^2(0*pi/180)-6)/sqrt(140)))+139233/35^3.5
0.00000: cos( 8*arccos((tan^2(0*pi/180)-7)/sqrt(224)))+sqrt(201521955687/28^8)
0.00000: cos( 9*arccos((tan^2(0*pi/180)-8)/sqrt(336)))-710734/21^4.5
0.00000: cos( 10*arccos((tan^2(0*pi/180)-9)/sqrt(480)))-sqrt(131608885690262651600/120^10)
0.00000: cos( 11*arccos((tan^2(0*pi/180)-10)/sqrt(660)))+sqrt(45708441953396645/33^11)
0.00000: cos( 12*arccos((tan^2(0*pi/180)-11)/sqrt(880)))+sqrt(301755509956546879568503440/220^12)
0.00000: cos( 13*arccos((tan^2(0*pi/180)-12)/sqrt(1144)))-sqrt(10456002758534337481188020418/143^13)
5.00000: cos( 9*arccos((tan^2(5*pi/180)-17)/sqrt(3264)))+sqrt(27105787751689263063202280/816^9)
7.00000: cos( 45*arccos((tan^2(7*pi/180)-89)/sqrt(469920)))-sqrt(239516470018385399663929538060557405217948365722536532529903795695859438482287750423866763936290150414849144979484446395225949498733648450715094430646288138299266341329903904192749863314857115036036302045835289175723008060796003/117480^45)
8.00000: cos( 15*arccos((tan^2(8*pi/180)-59)/sqrt(82880)))-sqrt(120892086374164617117804889503699181692921776309290071305586344/20720^15)
10.00000: cos( 6*arccos((tan^2(10*pi/180)-23)/sqrt(4640)))-sqrt(545350592228043043/1160^6)
11.00000: cos( 45*arccos((tan^2(11*pi/180)-89)/sqrt(469920)))-sqrt(241108705063247575295405211008663018888787388952077555582346745637718908113451774659441888335410056605861576508983876025586133837279045537275713777233684030935619593727690939031192117925171394965744440521651862991897021524451267/117480^45)
13.00000: cos( 45*arccos((tan^2(13*pi/180)-89)/sqrt(469920)))-sqrt(242199100225669933122792926333866890062343583749227470024790228954839607176512033233229226608245132277927265768536969285313974015962540756789515409529434810157789769656492887462879990157176378953864578885973788216381445310197773/117480^45)
14.00000: cos( 30*arccos((tan^2(14*pi/180)-119)/sqrt(691360)))-sqrt(2103884186820828843211577118575558188410177147596038757851409944913197255289731196858055918868836279294951669008097227097839901299198276690695034909978050937/172840^30)
16.00000: cos( 15*arccos((tan^2(16*pi/180)-59)/sqrt(82880)))-sqrt(138744831461068866637692610502523593093816995928609193743027604/20720^15)
22.00000: cos( 30*arccos((tan^2(22*pi/180)-119)/sqrt(691360)))-sqrt(2140043538053174990152356129582303351181495041236110439882297578152034047669884206161553319818487872939607202559729394143353493299793944960408685059571514759/172840^30)
25.00000: cos( 9*arccos((tan^2(25*pi/180)-17)/sqrt(3264)))+sqrt(31388360635585869027779359/816^9)
Vinicius Claudino Ferraz
2020-11-26 17:29:53 UTC
==> PRINT IF AND ONLY IF: Mod z = (b/2)^p
tan Arg z = Im z / Re z
|tan^2(m)-a)/b| <= 1 or complex parameters
|m| < 1e50
0 < |b|² < 1e50

Below pi - sqrt pi rad ; p = 3 sqrt pi

delta = 78.4458749614038683493456987190609345480708867207802057680645700921
m := delta*pi/180 = 1.36913880268427721116447589993835770139961994325271869276113680246
a = 17.0470201854089046173054131063059355505570749018890675676335121309
b = 46.4326573001899212329908198411443911283243933462199643143522956275
Re z = 5272055.42261976466404665292826975803006329902241059912422877213832
Im z = 17522685.8272321909952625964440552921876689139819283869015630515363
Mod z = 18298608.8809794051800912354972445969253486613354651831460313119388
cos Arg z = 0.288112361814554829942742491900146608330142614982156746490833965101 = [fraction] beta/gamma^p sometimes
sin Arg z = 0.957596609731696495785944558810968933211373845146420018468197755872
tan Arg z = 3.32369150598286036865991335105779592318107171124242841979519883454
==> cos( 5.31736155271654808189450245002343554839264836836716138464142336956 * arccos((tan^2(m)-a)/b)) = cos Arg z

Below 7 + 11j ; p = 3 + 4j ; j^2 = -1

delta = (7.0 + 11.0j)
m := delta*pi/180 = (0.122173047639603070384658353794202889941001032197920781926803400812
+ 0.191986217719376253461605984533747398478715907739589800170691058419j)
a = (0.936950669428743546827768846929707504774870887151232736096528249593
+ 0.561488752033197850462417808819696244872385560886434014812712673562j)
b = (5.56170422623450207757040405792579997985112645581611548259136122675
- 3.4582206135460884350097040989146913421932289031226128933270778982j)
Re z = (48816.8471587444468599046814485062822046439787619980023107963533663
- 179788.976454452513685412551796711841441165948972506627258637983057j)
Im z = (179789.237897823043124125744002993760997622469749352174795140305006
+ 48816.7376995858872943112458875837361736377356578678608062496646171j)
Mod z = (-324.270267255869437426878637642388438356399361586350376925121461125
+ 21.3301629442977368028019046907420725995973136225242727572235991904j)
cos Arg z = (-186.208608731826815717539847805837893772968988910740106757289092005
+ 542.193146403524594515388230539156565865211503917320998823535725525j) = [fraction] beta/gamma^p sometimes
sin Arg z = (-542.193971289829619842804099494340477021394621500280400829866630415
- 186.208325436660594483017010067557065238678905875727955893322708061j)
tan Arg z = (0.000000934746333617013429873604716164959686513941824526187920337371874295
+ 1.00000120036281358848344703283687594712462521855487887866326149364j)
==> cos( (3.0 + 4.0j) * arccos((tan^2(m)-a)/b)) = cos Arg z
mitchr...@gmail.com
2020-11-26 18:49:39 UTC
Post by ***@gmail.com
A zero angle in a trig table doesn't belong.
No polygon can have a zero angle...
They need to make a correction
and give an infinitesimal angle
A zero angle does not belong but
first quantity infinitely small does...

Mitchell Raemsch
Vinicius Claudino Ferraz
2020-11-27 23:23:29 UTC
Post by ***@gmail.com
A zero angle does not belong but
first quantity infinitely small does...
Mitchell Raemsch
And you? Are you conscious or unconscious?

I have divided 30 degrees into 180 equal parts.

cos(180*arccos((tan^2(1/6*pi/180)-239)/sqrt(10253120)))-sqrt(15503372195347364636534922587243368862857132839154923959108862842294544584011896059744252486728274410794463274185955701481429013721550326339714535486601864008848247513950002899195323068034685863346574474767161403854236117360590334016977088915028708816152632008188450808947948642600752608197643242981341975473895031328606045274692620564670307412508294281275611839175059955291574229219072553849958146087220524896967147325648572024724137188144841011156831056643081857820203444377438471699344374437353555496206475227353956725271424761684349651261276467675129856254333574718306940760806673580547259223259590540850945355685900978644395070926090471192418322853493487761970338125510901822106529590972276915402570465531267943504201936498138898714541401170680301128681219456018812708320665604257628758935413941266756811880788229411463040449249862262809877639353710235939941177295601095858507330449392535141922697976665475104032588947927137004353671355034396428798908137966029248120720811561700741469586573455122155370408700066630533630766944846979581230425235042338152185210812464680708486681883323437577553238852508058051323046467793077180194996890805776620385997/2563280^180)

But my computer has its memory limits.

Quarentine's motivation: I was working with integers sqrt(a/b)

today I started to detect a + sqrt(b) = 6.4142135...
it's almost a diophantine equation.

With that is easy to detect a - sqrt(b) = 4,585786437...

But I think I am far from -3 + sqrt(101)/7

You know 4 cos 15° = sqrt(6) + sqrt(2). It's only a printing problem. DCC has an area called data visualisation...
Chris M. Thomasson
2020-11-28 03:56:25 UTC
Post by Vinicius Claudino Ferraz
Post by ***@gmail.com
A zero angle does not belong but
first quantity infinitely small does...
Mitchell Raemsch
And you? Are you conscious or unconscious?
I have divided 30 degrees into 180 equal parts.
0.1666... degrees
Post by Vinicius Claudino Ferraz
cos(180*arccos((tan^2(1/6*pi/180)-239)/sqrt(10253120)))-sqrt(15503372195347364636534922587243368862857132839154923959108862842294544584011896059744252486728274410794463274185955701481429013721550326339714535486601864008848247513950002899195323068034685863346574474767161403854236117360590334016977088915028708816152632008188450808947948642600752608197643242981341975473895031328606045274692620564670307412508294281275611839175059955291574229219072553849958146087220524896967147325648572024724137188144841011156831056643081857820203444377438471699344374437353555496206475227353956725271424761684349651261276467675129856254333574718306940760806673580547259223259590540850945355685900978644395070926090471192418322853493487761970338125510901822106529590972276915402570465531267943504201936498138898714541401170680301128681219456018812708320665604257628758935413941266756811880788229411463040449249862262809877639353710235939941177295601095858507330449392535141922697976665475104032588947927137004353671355034396428798908137966029248120720811561700741469586573455122155370408700066630533630766944846979581230425235042338152185210812464680708486681883323437577553238852508058051323046467793077180194996890805776620385997/2563280^180)
But my computer has its memory limits.
Quarentine's motivation: I was working with integers sqrt(a/b)
today I started to detect a + sqrt(b) = 6.4142135...
it's almost a diophantine equation.
With that is easy to detect a - sqrt(b) = 4,585786437...
But I think I am far from -3 + sqrt(101)/7
You know 4 cos 15° = sqrt(6) + sqrt(2). It's only a printing problem. DCC has a
Vinicius Claudino Ferraz
2020-11-29 19:22:38 UTC
Today I'm prepared to amplify my huge list.

I was playing with periodic continued fractions.

a_1 = 5 + 1/ 1
a_ 2 = 1 + 1/a_ 1
a_ 3 = 10 + 1/a_ 2
a_ 4 = 5 + 1/a_ 3
a_ 5 = 2 + 1/a_ 4
a_ 6 = 1 + 1/a_ 5
a_ 7 = 3 + 1/a_ 6
x = 0 + 1/a_ 7
x = 1237 / 4561
True ; 0 ; [ 0. 3. 1. 2. 5. 10. 1. 5. 1.]
flag, fixed, vec = detect_sqrt(3 + mp.sqrt(2)/5)
a_1 = 6 + 1/x
a_ 2 = 1 + 1/a_ 1
x = 1 + 1/a_ 2
a_ 3 = 3 + 1/x
y = 3 + 1/a_ 3
7 x^2 + -12 x + -2 = 0
x_1 = ( 12 + 1 sqrt( 200 ))/( 14 + 0 sqrt( 200 ))
x_2 = ( 12 - 1 sqrt( 200 ))/( 14 + 0 sqrt( 200 ))
y_1 = ( 162 + 10 sqrt( 200 ))/( 50 + 3 sqrt( 200 ))
y_2 = ( 162 - 10 sqrt( 200 ))/( 50 - 3 sqrt( 200 ))
True ; 2 ; [3. 3. 1. 1. 6.]
flag, fixed, vec = detect_sqrt((-5 - mp.sqrt(7))/3)
a_1 = 1 + 1/x
a_ 2 = 1 + 1/a_ 1
a_ 3 = 1 + 1/a_ 2
x = 4 + 1/a_ 3
a_ 4 = 2 + 1/x
y = -3 + 1/a_ 4
3 x^2 + -12 x + -9 = 0
x_1 = ( 12 + 1 sqrt( 252 ))/( 6 + 0 sqrt( 252 ))
x_2 = ( 12 - 1 sqrt( 252 ))/( 6 + 0 sqrt( 252 ))
y_1 = ( -78 + -5 sqrt( 252 ))/( 30 + 2 sqrt( 252 ))
y_2 = ( -78 - -5 sqrt( 252 ))/( 30 - 2 sqrt( 252 ))
True ; 2 ; [-3. 2. 4. 1. 1. 1.]

Below cos^2 15 degree
a_1 = 12 + 1/x
x = 1 + 1/a_1
a_2 = 13 + 1/x
a_3 = 1 + 1/a_2
y = 1/a_3
12 x^2 -12 x -1 = 0
x_1 = (12 + sqrt(192))/24
x_2 = (12 - sqrt(192))/24
y_1 = (180 + 13 sqrt(192))/(192 + 14 sqrt(192))
y_2 = (180 - 13 sqrt(192))/(192 - 14 sqrt(192))
True ; 3 ; [ 0. 1. 13. 1. 12.]

Below cos^2 18 degree
a_1 = 8 + 1/x
x = 2 + 1/a_1
a_2 = 9 + 1/x
a_3 = 1 + 1/a_2
y = 1/a_3
8 x^2 -16 x -2 = 0
x_1 = (16 + sqrt(320))/16
x_2 = (16 - sqrt(320))/16
y_1 = (160 + 9 sqrt(320))/(176 + 10 sqrt(320))
y_2 = (160 - 9 sqrt(320))/(176 - 10 sqrt(320))
True ; 3 ; [0. 1. 9. 2. 8.]

Below cos^2 36 degree
a_1 = 2 + 1/x
x = 8 + 1/a_1
a_2 = 1 + 1/x
a_3 = 1 + 1/a_2
a_4 = 1 + 1/a_3
y = 1/a_4
2 x^2 -16 x -8 = 0
x_1 = (16 + sqrt(320))/4
x_2 = (16 - sqrt(320))/4
y_1 = (36 + 2 sqrt(320))/(56 + 3 sqrt(320))
y_2 = (36 - 2 sqrt(320))/(56 - 3 sqrt(320))
True ; 4 ; [0. 1. 1. 1. 8. 2.]

Below cos^2 54 degree
a_1 = 2 + 1/x
x = 8 + 1/a_1
a_2 = 1 + 1/x
a_3 = 2 + 1/a_2
y = 1/a_3
2 x^2 -16 x -8 = 0
x_1 = (16 + sqrt(320))/4
x_2 = (16 - sqrt(320))/4
y_1 = (20 + sqrt(320))/(56 + 3 sqrt(320))
y_2 = (20 - sqrt(320))/(56 - 3 sqrt(320))
True ; 3 ; [0. 2. 1. 8. 2.]

Below cos^2 72 degree
a_1 = 8 + 1/x
x = 2 + 1/a_1
a_2 = 10 + 1/x
y = 1/a_2
8 x^2 -16 x -2 = 0
x_1 = (16 + sqrt(320))/16
x_2 = (16 - sqrt(320))/16
y_1 = (16 + sqrt(320))/(176 + 10 sqrt(320))
y_2 = (16 - sqrt(320))/(176 - 10 sqrt(320))
True ; 2 ; [ 0. 10. 2. 8.]

Below cos^2 75 degree
a_1 = 12 + 1/x
x = 1 + 1/a_1
a_2 = 14 + 1/x
y = 1/a_2
12 x^2 -12 x -1 = 0
x_1 = (12 + sqrt(192))/24
x_2 = (12 - sqrt(192))/24
y_1 = (12 + sqrt(192))/(192 + 14 sqrt(192))
y_2 = (12 - sqrt(192))/(192 - 14 sqrt(192))
True ; 2 ; [ 0. 14. 1. 12.]
Vinicius Claudino Ferraz
2020-11-29 23:00:32 UTC
Destiny's Irony.
In my free time, I was playing with division of an angle by p = 3 initially, now p is any complex number.
Related to that division, suddenly appeared z^(1/p) = A + Bi in the space M = {A, B are constructible with ruler and compass}.
Just for illustration, I have let the space S_1 = {square roots of fractions} ⊂ M.
Now, I'm going to S_2 = {square roots of x = a + sqrt(b)/c}.
I hope some professional guy reads me someday... Because M is too big for me.
cos 3 degrees ∈ S_3 = {ab + cd ; each one of a,b,c,d is in S_2}.

integers, fractions p/q ∈ M
if x ∈ M then sqrt(x) ∈ M
if x, y ∈ M then x + y, x - y, xy ∈ M
if x, y ∈ M, y ≠ 0 then x/y ∈ M

θ radians is a constructible angle if and only if ∃ x, y ∈ M, x ≠ 0 such that θ = arctan(y/x)
Therefore θ is in the dual space D = arctan(M).
Until now, nov/29th, 19:48 GMT -3, I am using arctan(sqrt(algebraic fraction( tan (p * input) ))).
Where sqrt(fraction) came from Cardano's discriminant.

I would like to extend to
arctan( M-like function( tan (p * input ) ) )
many things are more generic than the sqrt of a polynomial deg i divided by a polynomial deg j.
what is a polynomial degree pi? series? haha

Destiny's Irony. It is working, most times. I'm like that guy

I will never prove that.
Vinicius Claudino Ferraz
2020-11-29 23:16:56 UTC

As it's Sunday, I solved my own exercise 11.1 below's page #15

One year ago, I hadn't neither R nor matlab nor GrandPython for work.

#GreatGrandPython ,

Vinicius
Vinicius Claudino Ferraz
2020-11-30 20:24:14 UTC
Illustration:

https://pbs.twimg.com/media/EoF6-tBWEAEWLXf?format=png&name=900x900

I have no brother. I have a sister from 1975, but she doesn't help. Diane never sent me a single SMS or email.
well, it's my life. in this situation, I may even define the number brotherhood haha.

[I defy you to] Prove that for any int p, for any real angle divided by p,

-1 <= (m^2 - a)/b <= 1.

So the arcccos of it makes sense, without going to immaginary numbers.
Mathin3D
2020-12-01 03:15:47 UTC