Solve the following equations algebrically where 0 lessthano

Solve the following equations algebrically where 0 lessthanorequalto x lessthanorequalto 2pi. Give answer as exact solutions. 2sin^2x + sinx = 0 2 cos^2x + 3cosx + 1 = 0

Solution

1. 2sin^2 x + sin x = 0

2 sin^ 2 x = -sin x

2 sin x = -1

sin x = - 1/2

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

x = (2 pi - pi/6) = 11pi /6

b) 2 cos ^ 2 x + 3 cos x +1 = 0

2 cos ^2 x + 2 cos x + cos x+ 1=0

2 cos x ( cos x + 1) + 1( cos x +1) = 0

(cos x+1) ( 2 cos x + 1) = 0

cos x +1 = 0

cos x = -1

2 cos x +1 =0

cos x = -1/2

x = pi , 2pi/3