Solve the equation for x if 0 ≤ x < 2π. Give your answer in radians using exact values only. (Enter your answers as a comma-separated list.) 2 cos2 x − sin x − 1 = 0

Question

Solve the equation for x if 0 ≤ x < 2π. Give your answer in radians using exact values only. (Enter your answers as a comma-separated list.)

2 cos² x − sin x − 1 = 0

Answer 1

Use the identity cos^2 = 1-sin^2 so

2(1-sin^2x) -sinx -1 =0

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

-2sin^2 x -sinx +1 =0 ( multiply both sides by -1) gives you

2 sin^2 x +sin x -1 =0 ( factor this quadratic to )

( 2sin x -1 ) ( sin x + 1) = 0

2sin x = 1 sin x = -1

sin x = 1/2

x= pi/6, 5pi/6 x= 3pi/2

So final answer is x= pi/6, 5pi/6 and 3pi/2