Trigonometry and Trigonometric Equations
This domain tests identities, the solution of trigonometric equations over a stated interval, and the geometric use of the sine and cosine rules. Papers often chain parts: an identity proved in part (a) reduces an equation in part (b) to a single trigonometric ratio, so the earlier result is meant to be used.
Core rules
- Compound angles: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B).
- Double angle: sin 2A = 2 sin A cos A; cos 2A = cos^2 A - sin^2 A = 1 - 2sin^2 A = 2cos^2 A - 1; tan 2A = 2t/(1 - t^2).
- t-formula with t = tan(θ/2): sin θ = 2t/(1 + t^2), cos θ = (1 - t^2)/(1 + t^2).
- Harmonic form: a cos θ + b sin θ = R cos(θ - α), where R = sqrt(a^2 + b^2) and tan α = b/a.
- Triangles: a/sin A = b/sin B = c/sin C; a^2 = b^2 + c^2 - 2bc cos A; area = (1/2)ab sin C.
Worked example
Solve 2sin^2 θ + sin θ - 1 = 0 for 0 ≤ θ ≤ 2π. Factorise: (2 sin θ - 1)(sin θ + 1) = 0, so sin θ = 1/2 or sin θ = -1. The first gives θ = π/6 and θ = 5π/6; the second gives θ = 3π/2. The solution set is {π/6, 5π/6, 3π/2}.
Traps
- Dividing through by cos θ or sin θ and thereby discarding the solutions where that factor vanishes.
- Quoting a general solution such as θ = nπ + (-1)^n (π/6) without restricting it to the interval demanded.
- Mixing degrees and radians within one answer, or giving a negative angle when 0 ≤ θ ≤ 2π is specified.
- Using the sine rule in the ambiguous SSA case and reporting only the acute angle.