Trigonometry
Trigonometry in this paper covers radian measure, the fundamental identities, compound and multiple angles, transformation formulae, general solutions of trigonometric equations, the sine and cosine rules, and the basic inverse trigonometric functions.
Identities and formulae
sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ. For compound angles, sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. Doubling gives sin 2A = 2 sin A cos A and cos 2A = 1 − 2sin²A = 2cos²A − 1. The product-to-sum forms are worth memorising: 2 sin A cos B = sin(A+B) + sin(A−B), 2 cos A cos B = cos(A+B) + cos(A−B), 2 sin A sin B = cos(A−B) − cos(A+B).
Equations and triangles
If sin θ = k with principal value α, the general solution is θ = nπ + (−1)ⁿα. For cos θ = k it is θ = 2nπ ± α; for tan θ = k it is θ = nπ + α, with n an integer throughout. In a triangle, a/sin A = b/sin B = c/sin C = 2R and a² = b² + c² − 2bc cos A.
Worked example
Solve sin θ = 1/2. The principal value is α = π/6, so θ = nπ + (−1)ⁿ(π/6). Checking n = 0 gives π/6 and n = 1 gives π − π/6 = 5π/6, both of which satisfy the equation.
Traps
- Mixing degrees and radians within a single calculation.
- Squaring an equation and keeping the extraneous roots it introduces.
- Giving only the principal value when the general solution is asked for.
- Using sin⁻¹ x outside −1 ≤ x ≤ 1.