3sec^2x - 7secx + 2 = 0

Question

Solve, for 0° ≤ x ≤ 360° , for the following equation by expressing it in the factor term

Answer 1

let w = sec(x), then re-write the equation as 3w² - 7w + 2. Now factor the quadratic.

(3w -1)(w - 2) = 0 then set each piece equal to 0

3w - 1 = 0 or w = 1/3 and w - 2 = 0 or w = 2

Now plug the sec(x) back in for w to get sec(x) = 1/3 and sec(x) = 2

sec(x) cannot = 1/3 (the range is < -1 and > 1 so that provides no solution

sec(x) = 2 is true when cos(x) = 1/2 which is at x = π/3 and 5π/3.