Find all solutions in 0 2 pi of the following equation 2 sin

Find all solutions in [0, 2 pi) of the following equation. 2 sin^2 theta = cos 2 theta sin x cos x = Squareroot 2/4

Solution

8. a. 2 sin²theta=cos 2 theta

          2sin²theta= 1-2sin²theta

          4sin²theta = 1

           sin²theta=1/4

            sin theta= +-1/2

           theta= sin^-1(+-1/2)

          theta= pi/6, 5pi/6, 7pi/6, 11pi/6

b. sin x cos x= sqrt2 /4

Multiply both sides by 2

2sinx cos x=sqrt2/2

sin 2x= sqrt2/2

2x= sin^-1(sqrt2 /2)

x= pi/8, 3pi/8, 9pi/8, 11pi/8