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

DifficultyAll Level 3 Level 4 Level 5

1
$\text{f}(x) = 12\sin x\cos x - 8\cos^2 x + 9$
(a) Express $\text{f}(x)$ in the form

$a\sin 2x + b\cos 2x + c$

where $a$ , $b$ and $c$ are integers to be found.

(3 marks)
(b) Using your answer to part (a), express $\text{f}(x)$ in the form

$R\sin(2x - \alpha) + c$

where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$

State the exact value of $R$ and give the value of $\alpha$ , in radians, to 3 decimal places.

(3 marks)
(c) Hence, or otherwise,

(i) write down the maximum value of $\text{f}(x)$

(ii) find the second smallest positive value of $x$ for which $\text{f}(x)$ takes this maximum value, giving your answer to 2 decimal places.

(3 marks)
Level 4 · 9 marksTry it
2
(a) Express $\sin^{2} x$ in terms of $\cos 2x$ . (1 mark)
(b) Hence find the exact value of $\sin^{2}\dfrac{\pi}{8}$ . (3 marks)
Level 4 · 4 marksTry it
3
Solve, for $0 \leqslant x \leqslant \pi$ , the equation

$\sin 2x = \tan x$

giving your answers in terms of $\pi$ . (5 marks)
Level 4 · 5 marksTry it
4
(a) Show that $(\sin\theta + \cos\theta)^2 \equiv 1 + \sin 2\theta$ . (2 marks)
(b) Hence solve the equation $(\sin\theta + \cos\theta)^2 = \frac{3}{2}$ for $0^\circ \le \theta < 360^\circ$ . (4 marks)
(c) Determine the greatest possible value of $\sin\theta + \cos\theta$ , justifying your answer. (2 marks)
Level 4 · 8 marksTry it
5
Solve, for $-180^\circ \leqslant \theta < 180^\circ$ , the equation

$4\cos 2\theta + 14\sin\theta = 7$

giving your answers to $1$ decimal place. (7 marks)
Level 4 · 7 marksTry it

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