Find ALL solutions of the interval [0, 2pi)

Question

Find ALL solutions of the interval [0, 2π)

cos 2x = √3/2

If there are some potential answers that won't fit, please say the reason.

Thank you

Answer 1

cos2x = (sqr3)/2

2x=arccos((sqr3)/2) = pi/6, 11pi/6, 13pi/6, 23pi/6, ...

x = pi/12, 11pi/12, 13pi/12, 23pi/12

= 15, 165, 195, 345 degrees

but cos2x > 0

cos30 >0, cos330>0, cos390>0, cos690=cos330>0

Answer 2

2x = cos^-1(√3/2)

2x = -π/6 + 2kπ or π/6 + 2kπ, k any integer

x = -π/12 + kπ or π/12 + kπ, k any integer

k=0: -π/12 , π/12

k=1: 11π/12, 13π/12

k=2: 23π/12, 25π/12

The lined-out solutions are outside the target interval.

Check out solutions on Desmos:

desmos.com/calculator/9guwipwqhd

Note: the only rejected solutions are those outside [0, 2π).