solve this equation on the interval [0,2pi)

Question

cos(2theta) +3 = 5 cos theta

Answer 1

Using the double angle identity cos(2θ) = 2cos²(θ) - 1 we can substitute to get:

(2cos²(θ) - 1) + 3 = 5cos(θ)

2cos²(θ) - 5cos(θ) + 2 = 0

Let w = cos(θ)

2w² - 5w + 2 = 0

(2w - 1)(w - 2) = 0

w = 1/2 or w = 2

Back substituting:

cos(θ) = 1/2 and cos(θ) = 2

We can throw out cos(θ) = 2 because cosine alternates between plus and minus 1.

So cos(θ) = 1/2 when θ = π/3 and 5π/3