help with trig math question

Question

solve sin theta +1 = cos 2 theta on the interval theta is > or = to 0 and < 2pi

Answer 1

let's use x for theta

sinx+1=cos2x

use the trig identity cos2x=1-2sin^2x

sinx+1=1-2sin^2x

get all terms on one side

-1+2sin^2x+sinx+1=0

-1 and 1 cancel

2sin^2x+sinx=0

factor

sinx(2sinx+1)=0

set each factor equal to 0

sinx=0

2sinx+1=0, 2sinx=-1, sinx=(-1/2)

so sinx=0 and sinx=(-1/2)

now look on the unit circle to find at what angle sinx=0 and at what angle sinx=(-1/2)

sinx=0 at x=0° and at x=180°

sinx=(-1/2) at x=210° and at x=330°