Question

Solve 8cos^2(x) − 6cos(x) + 1 = 0 for all solutions 0 ≤ x < 2π

Thank you!

Answer 1

x ∈ {1.047198, 1.318116, 4.965069, 5.235988}

Explanation:

To find the values of cos(x), you solve this like you would any other quadratic equation. Factoring gives ...

(4cos(x) -1)(2cos(x) -1) = 0

which has solutions that make these factors zero:

cos(x) = 1/4

cos(x) = 1/2

Using the inverse cosine function, we have ...

x = ±arccos(1/4) +2kπ

x = ±arccos(1/2) +2kπ = ±π/3 +2kπ

The four smallest positive solutions will be the positive values with k=0 and the negative values with k=1.

x ∈ {1.047198, 1.318116, 4.965069, 5.235988}