Solving Trig Equations Step by Step

Pre-Calculus · Axiom Academy

EXAMPLE Solving Trigonometric Equations Solve a quadratic-in-sine, then read every solution off the unit circle. Solve for all values of x in : . It looks messy, but it's secretly a quadratic — substitute and the structure pops out. Three solutions in one full turn: one where sin x = 1 (top), two where sin x = -1/2 (Quadrants III and IV). Nice work — you turned a trig equation into a quadratic, solved it, and recovered every angle on the interval. Spot the quadratic: if a trig function appears squared and to the first power, substitute and factor like any quadratic. Back-substitute, don't stop at u: and u = 1 are not the answer — you still have to solve on the interval. Use the reference angle and signs: has reference angle and is negative in Quadrants III and IV, giving and . Result: — three solutions on . Always verify each by plugging back in. For the general solution, add to each (for any integer n ) — the pattern repeats every full turn.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.