Is there an identity for the cos( 1/3 x ) ?

[swampwiz]

It seems that this term comes up in solving the cubic equation. While there is the identity for the half-angle, there doesn't seem to be one for third-angle.

 

[coolul007]

http://en.wikipedia.org/wiki/Trig_Identity

 

[JJacquelin]

Hi !

The identity involves complex terms :

 

[swampwiz]

OK, I understand the general form. I didn't think to use an integer fraction in lieu of an integer. (Actually, it appears that *any* number could be used there.)

 

[CellCoree]

I can confirm that there is no specific identity for the cos(1/3x) term. However, this term does have significance in solving cubic equations, specifically in the process of applying trigonometric substitutions to solve them. While there is an identity for the half-angle (cos^2(x/2) = (1+cosx)/2), there is no specific identity for the third-angle. This is because the use of trigonometric substitutions in solving cubic equations involves manipulating various trigonometric identities and equations, rather than relying on a specific identity for the cos(1/3x) term. Therefore, while there may not be a specific identity for this term, it still plays a crucial role in solving cubic equations through the use of trigonometric substitutions.