Find all solutions in the Interval 0 2 pi theta pi4 7 pi4 U

Find all solutions in the Interval [0, 2 pi]. theta = pi/4, 7 pi/4 Use a Double- or Half-Angle Formula to solve the equation in sin 2 theta + sin theta = 0 theta = pi/2, 3 pi/2, 7 pi/6, 11 pi/6

Solution

Cos theta cos 2theta+sin theta sin 2theta=1/2

Cos(2theta-theta)=1/2

Cos theta=1/2

Theta=cos^-1(1/2)

Theta= pi/3, 5pi/3

Second question

sin2theta+sin theta=0

2sin(3theta/2) cos(theta/2)=0

Sin(3theta/2)=0 ,cos (theta/2)=0

3theta/2=0,pi,2pi theta/2=pi/2

Theta= 0, 2pi/3,4pi/3, pi