Trigonometric identities & equations
The idea
A trigonometric identity is an equation true for every allowed angle — not a puzzle to solve but a rewriting rule. The bedrock is sin²θ + cos²θ = 1, which is the Pythagorean theorem read off the unit circle, joined by quotient identities like tan θ = sin θ/cos θ and the angle-sum formulas. Identities matter because trig equations rarely arrive in solvable form; the craft lies in converting a mixed expression into a single function of a single angle so that ordinary algebra can take over.
Solving a trigonometric equation runs in two phases. Phase one is algebra: use identities to reach one trig function, then treat it like a variable — factor, isolate, apply the quadratic playbook. Phase two is geometry: a value like sin x = −1/2 corresponds to specific unit-circle points, and you must list every angle in the requested interval that reaches them. The recurring trap is stopping at the calculator's single inverse-function answer; an equation typically has several solutions per period, and the rest are silently discarded unless you sketch the circle and harvest them all.
Worked example
Solve 2cos²x + sin x − 1 = 0 for all x in the interval [0, 2π).
- Two different functions appear, so unify them with the Pythagorean identity: cos²x = 1 − sin²x turns the equation into 2(1 − sin²x) + sin x − 1 = 0.
- Expand and tidy: 2 − 2sin²x + sin x − 1 = 0 becomes −2sin²x + sin x + 1 = 0; multiplying by −1 gives 2sin²x − sin x − 1 = 0, a quadratic in s = sin x.
- Factor the quadratic: 2s² − s − 1 = (2s + 1)(s − 1) = 0, so sin x = −1/2 or sin x = 1.
- Read the circle: sin x = 1 happens only at x = π/2, while sin x = −1/2 happens at the quadrant III and IV angles with reference π/6, namely x = 7π/6 and x = 11π/6.
- Spot-check x = 7π/6: cos 7π/6 = −√3/2, so 2cos²x = 2 × 3/4 = 3/2, and 3/2 + (−1/2) − 1 = 0. The solution checks out in the original equation.
Answer. The solutions in [0, 2π) are x = π/2, x = 7π/6, and x = 11π/6.
Check your understanding
- In what sense is sin²θ + cos²θ = 1 the Pythagorean theorem in disguise, and where is the right triangle hiding?
- Why does a single value of sin x usually correspond to two angles in each period rather than one?
- When an equation mixes sine and cosine, what guides your choice of which one to eliminate?
- How would the solution set change if the interval [0, 2π) were replaced by all real numbers?
Build the foundations first
Trigonometric identities & equations builds on these concepts. If any feel shaky, start there.