Trigonometry
Trigonometry in the IPE papers tests transformations (sum-to-product and product-to-sum), general solutions of trigonometric equations, inverse trigonometric functions, hyperbolic functions, and the properties of triangles with sine rule, cosine rule, projection, tangent rule, half-angle formulae and radii.
Core rules and concepts
- sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2); the companion forms for sin C − sin D and cos C ± cos D are examined together.
- General solutions: sin θ = sin α gives θ = nπ + (−1)^n α; cos θ = cos α gives θ = 2nπ ± α; tan θ = tan α gives θ = nπ + α.
- Principal ranges: sin⁻¹x ∈ [−π/2, π/2], cos⁻¹x ∈ [0, π], tan⁻¹x ∈ (−π/2, π/2). For tan⁻¹x + tan⁻¹y, add π when xy > 1.
- a cos θ + b sin θ lies in [−√(a²+b²), √(a²+b²)], attained at the auxiliary angle.
- In triangle ABC: a/sin A = b/sin B = c/sin C = 2R; a² = b² + c² − 2bc cos A; Δ = abc/4R = rs.
- Hyperbolic: cosh²x − sinh²x = 1.
Worked example
Solve 2sin²θ − 3sin θ + 1 = 0. Factorise: (2sin θ − 1)(sin θ − 1) = 0, so sin θ = 1/2 or sin θ = 1. sin θ = 1/2 gives θ = nπ + (−1)^n (π/6); sin θ = 1 gives θ = nπ + (−1)^n (π/2). Both families must appear in the answer.
Traps that cost marks
- Dividing through by sin θ or cos θ and silently losing roots.
- Giving a general solution in degrees while the paper works in radians.
- Using nπ where 2nπ is required, or dropping the (−1)^n.
- Ignoring the quadrant adjustment in inverse tangent addition.