Find all solutions of the equation in the interval 0 2pi cos

Find all solutions of the equation in the interval [0, 2pi]. cos2x+ sin x = 0 Write your answer in radians in terms of pi. If there is more than one solution, separate them with commas.

Solution

cos2x +sinx =0 ; interval [0, 2pi)

use the formula : cos2x = 1-2sin^2x

1-2sin^2x +sinx =0

solve the quadratice equation:

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

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

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

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

sinx = -1/2

x = pi +pi/6 , 2pi -pi/6 = 7pi/6 , 11pi/6

sinx = 1

x = pi/2

Solution : x = pi/2 , 7pi/6 , 11pi/6