Question

How do you solve `2cos^(2)x - cosx = 1`?

Answer 1

Call cos x = t. Solve quadratic equation:

f(t) = 2t^2 - t - 1 = 0

Since (a + b + c = 0) one real root is t = 1 and the other is t = c/a = -1/2.

Use the trig unit circle as proof to solve:

a. cos x = t = 1 -> x 0, and x = 2pi

b. cos x = -1/2 -> x = 2pi/3, and x = 5pi/3

Within the interval (0, 2pi), there are 4 answers: `0, (2pi)/3, (5pi)/3, and 2pi`
Check:
`x = 2pi -> cos x = 1 -> f(x) = 2 - 1 = 1`Correct
`x = (2pi)/3 -> cos x = -1/2 -> f(x) = 1/2 + 1/2 = 1` Correct