# Exact value of sin 65

1. Nov 12, 2015

### songoku

1. The problem statement, all variables and given/known data
Find the exact value of sin 65o

2. Relevant equations
Trigonometry Identities

3. The attempt at a solution
I tried using sin 3x = 3 sin x - 4 sin3x but ended with nasty algebra.

sin (3 . 65o) = 3 sin 65 - 4 sin3 65o
sin (195o) = 3 sin 65 - 4 sin3 65o
sin (180o+15o) = 3 sin 65 - 4 sin3 65o
- sin (15o) = 3 sin 65 - 4 sin3 65o ; let sin 65o = x
(√2 - √6) / 4 = 3x - 4x3

Can't solve for x

Thanks

2. Nov 12, 2015

### Staff: Mentor

There is a formula for analytic solutions for the last equation. Using that is probably easier than finding different trigonometric identities.

You could try to establish a value for sin(10o) first I guess.

3. Nov 12, 2015

### Ray Vickson

Somebody has published (on the internet) a list of exact formulas involving roots and the like, for the sine function for all angles in 1 degree increments from 1 degree to 90 degrees. The work includes proofs of all the results. (It was done as a retirement project by an ex-professor of mathematics.) For more details, see

http://www.intmath.com/blog/mathematics/how-do-you-find-exact-values-for-the-sine-of-all-angles-6212

Last edited: Nov 12, 2015
4. Nov 14, 2015

### songoku

Thanks for all the help