Variable substitution and quadratic equations

Why use a change of variable?

Some trigonometric equations resemble quadratic equations, but with cos(x) or sin(x) in place of x. The method involves introducing a new variable to reduce the problem to one we are already familiar with.

Complete example

Solve 2cos²(x) - cos(x) - 1 = 0 on [0, 2π[.

Step 1: Let X = cos(x). The equation becomes 2*X² - X - 1 = 0.

Step 2: We solve this standard quadratic equation. Discriminant: δ = (-1)² - 4 × 2 × (-1) = 1 + 8 = 9. √δ = 3. X₁ = (1 − 3)/4 = −1/2 and X₂ = (1 + 3)/4 = 1.

Step 3: We return to x by solving cos(x) = -1/2 and cos(x) = 1 separately.

cos(x) = 1 -> x = 0 (the only solution in [0, 2π[).

cos(x) = -1/2 -> cos(x) = cos(2π/3), so x = 2π/3 + 2kπ or x = -2π/3 + 2kπ. On [0, 2π[: x = 2π/3 or x = 4π/3.

Set of solutions: x = 0, 2π/3, 4π/3.

Check that X is a valid value

As cos(x) and sin(x) are always between -1 and 1, we must always check that the values of X found lie within [-1, 1]. If a value of X lies outside this interval, it does not yield any solution for x, so we discard it.

Common pitfall

Forgetting to return to the variable x after solving the equation for X: finding X = 1 or X = -1/2 is not an answer to the question; these are merely intermediate steps.