Exact Expression for Sine of 1 Degree

[bob012345]

https://www.wolframalpha.com/input/?i=sin(pi/10)
S10=sin(π/10)=1/4(5−1)

S20=sin(π/20)=(1−1−S102)/2

Which gets you to sine of 9 degrees.
But then you would need to apply the one-third angle formula twice - and that involves complex roots.

https://www.quora.com/What-is-the-formula-for-sin-x-3-one-third-angle-formula
S60=sin(π/60)=((−S20+S202−1)1/3+(−S20−S202−1)1/3)/2
S180=sin(π/180)=((−S60+S602−1)1/3+(−S60−S602−1)1/3)/2

The Sine of $3°$ is known in closed form but when put in the third angle formula and solving the cubic, you get back to cube roots of complex numbers. I know how to solve that but it's too massive to typeset.

I found this work that shows how ugly the expressions are for those angles not multiples of $3°$ and they all involve cube roots of complex numbers.

https://www.intmath.com/blog/wp-content/images/2011/06/exact-values-sin-degrees.pdf

 

[.Scott]

The Sine of $3°$ is known in closed form but when put in the third angle formula and solving the cubic, you get back to cube roots of complex numbers. I know how to solve that but it's too massive to typeset.

I found this work that shows how ugly the expressions are for those angles not multiples of $3°$ and they all involve cube roots of complex numbers.

https://www.intmath.com/blog/wp-content/images/2011/06/exact-values-sin-degrees.pdf

If what you say is true, there are two closed formula ways of getting $sin(3°)$, one using the third angle formula (described in my post above), one that avoids roots of a complex number (as described by you). That suggests (strongly suggests) that there may be a method of performing the third angle formula without taking the root of a complex number. If that is the case, then there is a solution.

Clearly it would be ugly - but not so ugly if it was expressed as a progressive series of equations - as with the series that I presented. Values are commonly expressed as a combination of operations such as add, subtract, divide, square root. Expressing a value as a series of operations just means that you are expanding your available operations before providing the final formula. It's equivalent to $S_180 = f_a(f_b(f_c(f_d(f_e()))))$. A great way to keep the ugliness down.

 

[.Scott]

(edited in response to comment from @bob012345)
from this page (intmath.com), a solution that include the cube roots of complex numbers:

A comprehensive list is available here: Exact value from sine 1° to sine 90°

 

[bob012345]

from this page (intmath.com):
View attachment 269775
A comprehensive list is available here: Exact value from sine 1° to sine 90°

Yes, but my view is that this is not really the solution I sought. There is one real root which must be the Sine of 1° but expressing it in a closed form without roots of complex numbers or Sines and Cosines of other angles which was the original intent of the post, is not possible

 

[bob012345]

(edited in response to comment from @bob012345)
from this page (intmath.com), a solution that include the cube roots of complex numbers:
View attachment 269775
A comprehensive list is available here: Exact value from sine 1° to sine 90°

I tried evaluating the simpler case of Sin(20°) and after a boatload of computation I got this as one of 9 possibilities;

$$Sin(20°)= \frac 1 2( (Cos(10°) - \sqrt 3 Sin(10°))$$

Alternativly, I also got;

$$Sin(20°)= -\frac 1 2 ((Cos(50°) - \sqrt 3 Sin(50°))$$

Which I could have arrived at much easier by just using the identities;

$$Sin(a ± b) = Sin(a)Cos(b) ± Cos(a)Sin(b)$$

Essentially, evaluating these cubic roots of complex numbers either gives more cubic roots of complex numbers or if you use De Moivre's theorem you end up deriving several trigonometric identities. At least that is my experience.