Exact Value of Sin 65 Degrees: Trigonometry Identities for Finding the Solution

[songoku]

Homework Statement

Find the exact value of sin 65^o

Homework Equations

Trigonometry Identities

The Attempt at a Solution

I tried using sin 3x = 3 sin x - 4 sin³x but ended with nasty algebra.

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

Can't solve for x

Thanks

 

[mfb]

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(10^o) first I guess.

 

[Ray Vickson]

Homework Statement

Find the exact value of sin 65^o

Homework Equations

Trigonometry Identities

The Attempt at a Solution

I tried using sin 3x = 3 sin x - 4 sin³x but ended with nasty algebra.

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

Can't solve for x

Thanks

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

 

[songoku]

Thanks for all the help