Trigonometry
Core concepts
Know the unit circle, radian measure, the six trigonometric functions, exact values at 0, π/6, π/4, π/3, π/2, and the signs by quadrant (ASTC). Master the Pythagorean identities, double-angle and sum formulas, and solve equations like sin x = k and a sin²x + b sin x + c = 0. Understand graphs, amplitude, period, and phase shift of a sin(bx + c) + d.
Traps
- Forgetting that sin x = k has infinitely many solutions; the general solution must include the period 2πn.
- Sign errors when converting between quadrants, especially for tangent.
- Confusing degrees and radians in the same problem.
- Losing solutions by dividing both sides by sin x or cos x; factor instead.
- Misapplying sin(A + B) ≠ sin A + sin B.
Worked example
Solve 2sin²x − sin x − 1 = 0 on [0, 2π). Let t = sin x: 2t² − t − 1 = 0, so (2t + 1)(t − 1) = 0, giving t = −1/2 or t = 1. From sin x = 1: x = π/2. From sin x = −1/2: x = 7π/6 or 11π/6. The solution set is {π/2, 7π/6, 11π/6}.
Rule of thumb
When an equation mixes powers of one trig function, substitute t and solve the polynomial; then restore all solutions in the required interval.