Question

How do you find all solutions in the interval [0,2pi) of `cos 2x - sqrt 3 sin x=1`?

Answer 1

Only one solution for sin x = (`sqrt`19 - `sqrt`3) / 4.
Accordingly, x = 41.05 deg and x = 180 - 41,05 = 138.95 deg

Explanation:

cos 2x = 1 - 2 sin x. sin x. Solving the quadratic, we get one root as above, The other is < - 1 and is inadmissible. Also, arcsin gives the value in [0, `pi/2`], for positive sines. Use sin (`pi` - x) to get the second solution in [0, 2`pi`].