A-level · Trigonometry

Addition formulaeExam-style questions

The compound-angle formulae for sin(A±B), cos(A±B) and tan(A±B) — the sign patterns to get right, and using them for exact values of non-standard angles, expanding, proving identities and solving equations.

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

DifficultyAll Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Show that $\cos 3x$ can be written in the form

$P\cos^3 x + Q\cos x$

where $P$ and $Q$ are constants to be found.

(4 marks)
(b) Hence or otherwise, solve, for $0 < \theta \leqslant 360°$

$2\cos 3\theta + 3\cos 2\theta + 4 = 0$

giving your answers, in degrees, to one decimal place.

(4 marks)
Level 4 · 8 marksTry it
2
$A$ is acute with $\sin A = \frac{4}{5}$ , and $B$ is obtuse with $\sin B = \frac{5}{13}$ . Find the exact value of $\sin(A - B)$ .
Level 4 · 3 marksTry it
3
In this question you must show detailed reasoning.

Solve the equation

$\sin\left(2x - \dfrac{\pi}{6}\right) = \cos 2x$

for $0 \le x < 2\pi$ , giving your answers in exact form. (5 marks)
Level 4 · 5 marksTry it
4
In this question you must show detailed reasoning.
(a) Prove that

$\tan\left(\theta + \dfrac{\pi}{4}\right) + \tan\left(\theta - \dfrac{\pi}{4}\right) \equiv 2\tan 2\theta$

for all values of $\theta$ for which both sides are defined. (4 marks)
(b) Hence solve the equation

$\tan\left(\theta + \dfrac{\pi}{4}\right) + \tan\left(\theta - \dfrac{\pi}{4}\right) = -2\sqrt3$

for $0 \leqslant \theta < \pi$ , giving your answers in terms of $\pi$ . (4 marks)
Level 4 · 8 marksTry it
5
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
(a) Given that

$5\sin(x - 30^\circ) = 2\cos(x - 60^\circ)$

show that

$\tan x = \frac{7\sqrt{3}}{9}$

(4 marks)
(b) Hence or otherwise solve, for $0 \leqslant \theta < 180^\circ$ , the equation

$5\sin(2\theta - 30^\circ) = 2\cos(2\theta - 60^\circ)$

giving your answers to one decimal place.

(4 marks)
Level 4 · 8 marksTry it

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