Trigonometry
Core concepts
You must know the exact values, the graphs and symmetries of sine, cosine and tangent, radian measure, arc length and sector area, the compound-angle and double-angle identities, and the solution of trigonometric equations over a stated interval. Recognise and use identities such as sin²θ + cos²θ = 1 and the forms R sin(θ ± α), R cos(θ ± α) for combining a sin θ + b cos θ.
Traps
- Failing to find all solutions in the given interval. Use the symmetry of the graph or a CAST diagram to obtain the second and further solutions.
- When the argument is 2θ or θ + 30°, scale the interval before solving, then divide back.
- Squaring an equation can introduce extraneous solutions; always check.
- Radian mode versus degree mode on a calculator is a frequent source of wrong answers in calculus and arc-length questions.
Worked example
Solve 3 sin θ = 2 cos²θ for 0 ≤ θ < 360°. Use cos²θ = 1 − sin²θ: 3 sin θ = 2 − 2 sin²θ, so 2 sin²θ + 3 sin θ − 2 = 0. Factor: (2 sin θ − 1)(sin θ + 2) = 0. The second factor gives sin θ = −2 (impossible). So sin θ = ½, giving θ = 30° or 150°. Rule of thumb: reduce to a single trigonometric function before solving.