A-level · Trigonometry

Double-angle formulaeExam-style questions

The double-angle formulae for sin 2A, cos 2A (three forms) and tan 2A, where they come from, choosing the right form of cos 2A, and using them to solve equations and find exact values.

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23 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 3Level 4Level 5
  1. 1
    (a) Prove that

    cos⁡2θcos⁡θ+sin⁡θ≡cos⁡θ−sin⁡θ,cos⁡θ+sin⁡θ≠0\dfrac{\cos 2\theta}{\cos\theta + \sin\theta} \equiv \cos\theta - \sin\theta, \qquad \cos\theta + \sin\theta \neq 0

    (3 marks)
    (b) Hence solve, for 0⩽θ⩽2π0 \leqslant \theta \leqslant 2\pi, the equation

    cos⁡2θcos⁡θ+sin⁡θ=12\dfrac{\cos 2\theta}{\cos\theta + \sin\theta} = \dfrac12

    giving your answers to 33 decimal places. (5 marks)
    Level 5 · 8 marksTry it
  2. 2
    In this question you must show all stages of your working.

    Solutions relying entirely on calculator technology are not acceptable.
    (a) Show that the equation

    2sin⁡2θ1−cos⁡2θ+3cosec⁡2θ=11\frac{2\sin 2\theta}{1 - \cos 2\theta} + 3\operatorname{cosec}^2\theta = 11

    may be written as

    3cot⁡2θ+2cot⁡θ−8=03\cot^2\theta + 2\cot\theta - 8 = 0

    (4 marks)
    (b) Hence solve, for 180∘<x<540∘180^\circ < x < 540^\circ, the equation

    2sin⁡2x1−cos⁡2x+3cosec⁡2x=11x∈R, x≠360∘\frac{2\sin 2x}{1 - \cos 2x} + 3\operatorname{cosec}^2 x = 11 \qquad x \in \mathbb{R},\ x \neq 360^\circ

    giving your answers to one decimal place.

    (4 marks)
    Level 5 · 8 marksTry it