solve the equation on the interval 0≤θ<2π

Question

sin^2 theta -cos^2 theta +cos theta =0

Answer 1

sin²x - cos²x + cos x = 0; 0 ≤ x < 2π

-(cos²x - sin²x) + cos x = 0 rearrange 1st two terms

-(2cos²x - 1) + cos x = 0 apply double-angle formula

2cos²x - cos x - 1 = 0 multiply each term by -1 & rearrange to standard form

(2cos x + 1)(cos x - 1) = 0 factor the trinomial

2cos x = -1 or cos x = 1 apply Zero Product Property

cos x = -1/2 solve each equation

x = 2π/3, 4π/3

or cos x = 1

x = 0

Solutions: x = 0, 2π/3, 4π/3