find all solutions of the equation in the interval [0,2pi) leave answers in radians

Question

cos2θ+3=5cosθ

Answer 1

cos2x = cos^2x - sin^2x

sin^2x = 1-cos^2x

cos2x = cos^2x - (1-cos^2x) = cos^2x -1 + cos^2x = 2cos^2x -1

cos2x + 3 = 5cosx = 2cos^2x - 1 +3

2cos^2x -5cosx +2 = 0

factor

(2cosx-1)(cosx-2) = 0

cosx = 1/2 (cosx=2 is impossible, as cosx is never greater than 1)

x = 60 degrees or pi/3 or -120 degrees or 300 degrees or 5pi/3

theta = pi/3 or 5pi/3

both between 0 and 2pi