A-Level Maths: Trigonometry

A-Level trigonometry is identity-driven: the exam rewards candidates who can pick the right identity, apply it cleanly, and keep track of signs and quadrants. This quiz spans the full identity toolkit from the Pythagorean family to the R-form.

The identity questions open with the two derived Pythagorean identities — 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ — tested as recognitions against plausible corruptions. The reciprocal functions get a dedicated question on their ranges (|sec θ| ≥ 1, |cosec θ| ≥ 1), the property that catches students who treat them like ordinary functions.

Compound and double angles form the computational heart of the set: an exact-value calculation of sin(A + B) from cos A = 3/5 and sin B = 5/13 with both angles acute (requiring the missing sides of two right triangles before the formula applies, landing on 63/65), the three equivalent forms of cos 2θ tested together, tan(7π/12) built from the compound angle formula for π/3 + π/4 and rationalised to −2 − √3, and cos 2θ evaluated from sin θ = 1/3 through the 1 − 2sin²θ form. A simplification question reduces sin 2θ / (1 + cos 2θ) to tan θ — the double-angle collapse examiners love.

The R-form question converts 3cos θ + 4sin θ into R cos(θ − α), extracting R = 5 and tan α = 4/3, the setup step for solving equations and finding maxima. Equation-solving appears through sec θ = 2 over a full period, and radian fluency through the small-angle approximations sin θ ≈ θ and tan θ ≈ θ.

Every explanation names the identity used and shows the substitution, mirroring the "show that" working the written paper demands.

• Apply the Pythagorean identity family including sec and cosec forms
• Compute exact compound-angle values from given ratios
• Use all three forms of cos 2θ and simplify double-angle expressions
• Express a cos θ + b sin θ in R cos(θ − α) form
• Solve reciprocal trig equations and use small-angle approximations in radians

Topic scope follows section E (Trigonometry) of the DfE's prescribed AS and A level mathematics subject content — 100% common across all exam boards: identities, compound and double angle formulae, the R-form, trigonometric equations, radians, small angles, and reciprocal functions.