Trigonometry
Trigonometry in the Pancyprian Core Mathematics paper mixes exact values, identities and the solution of equations on a stated interval with triangle problems handled by the sine and cosine rules. Marks are lost mainly through missing solutions and through a calculator left in the wrong angle mode.
Angles, arcs and signs
- π radians = 180°; arc length s = rθ and sector area ½r²θ, with θ in radians.
- Signs follow CAST: all functions positive in quadrant I, sine in II, tangent in III, cosine in IV.
- Exact values at 0, π/6, π/4, π/3 and π/2 should be immediate: sin π/6 = ½, cos π/4 = √2/2, tan π/3 = √3.
Identities
- sin²x + cos²x = 1 and tan x = sin x / cos x, valid wherever cos x ≠ 0.
- Double angle: sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1.
Solving equations
- Reduce the equation to a single trigonometric function, factorise as a quadratic if needed, then produce every solution lying inside the given interval using the reference angle and symmetry.
Triangles
- Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A.
- Area of a triangle = ½ab sin C.
- With two sides and a non-included angle (SSA), the sine rule can yield two valid triangles - check both before discarding one.
Worked example
Solve 2sin²x + sin x − 1 = 0 for 0 ≤ x < 2π. Put u = sin x: 2u² + u − 1 = 0 factorises as (2u − 1)(u + 1) = 0, so u = ½ or u = −1. Now sin x = ½ gives x = π/6 and x = 5π/6; sin x = −1 gives x = 3π/2. The solution set is {π/6, 5π/6, 3π/2}.
Traps that cost marks
- Dividing through by sin x, which destroys the sin x = 0 family of solutions.
- Stopping at the first-quadrant angle and forgetting the symmetric solution in the second quadrant.
- Leaving the calculator in degree mode when the interval is given in radians.
- Sign slips in the term −2bc cos A when A is obtuse, where cos A is negative and the term becomes positive.