Maths: When a Trig Equation Is Really a Quadratic

Many trigonometric equations that look unfamiliar at first are really quadratics in disguise. When an equation contains a single trigonometric function raised to different powers — here, cos²x and cosx — substituting u = cosx turns it into a standard quadratic such as 2u² − 3u + 1 = 0. Factorising or applying the quadratic formula then reveals the possible values of cosx, which become the starting points for finding x itself. This matters because the domain shapes the answer. Each value of cosx typically yields two solutions within 0 ≤ x ≤ 2π, found using the symmetry of the cosine curve: if x is a solution, so is 2π − x. The boundary cases where cosx = 1 are the exception, each producing a single solution at the endpoints of the interval. Recognising how substitution, factorisation, and the periodic symmetry of cosine connect is the key skill — one equation, several solutions, all captured within one full revolution.