Question

How do you solve `2 + 3cos2x = cosx`?

Answer 1

Replace cos 2x by `(2cos^2 x - 1)`
f(x) = 2 + 3(2cos^2 x - 1) - cos x = 0. Call cos x = t.

6t^2 - t - 1 = 0. Solve by the formula
`D = d^2 = 1 + 24 = 25 -> d = +- 5`

`t = 1/12 + 5/12 = 6/12 = 1/2`

`t = 1/12 - 5/12 = -4/12 = -1/3`
Use the trig unit circle as proof:

`cos x = t = 1/2 -> x = pi/3 and x = 5pi/3`

`cos x = -1/3 -> x = 109.47 and x = 250.53 deg`