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

Find all solutions of the equation in the interval [0, 2 pi]. cos 2x+ cos x = -1 Write your answer in radians in terms of pi. If there is more than one solution, separate them with commas.

Solution

cos2x +cosx =-1 interval [ 0, 2pi )

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

2cos^2x - 1 +cosx = -1

2cos^2x +cosx =0

cosx( 2cosx +1) =0

cosx =0

x = pi/2 , 3pi/2

cosx = -1/2

x = pi-pi/3 , pi+pi/3 ( cos is -ve in IInd and IIIrd quadrant)

x = 2pi/3 , 4pi/3

Solution : x= pi/2 , 2pi/3, 3pi/2 , 4pi/3