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 detailed reasoning.
(a) (i) Starting from the identity for $\tan(A + B)$ , show that $\tan 3\theta \equiv \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}.$

(ii) Solve the equation $\tan 3\theta = 1$ for $0^\circ \le \theta \le 180^\circ$ .

(5 marks)
(b) (i) Expand $(x + 1)(x^2 - 4x + 1)$ .

(ii) Hence solve the equation $x^3 - 3x^2 - 3x + 1 = 0$ , giving each root in exact form.

(3 marks)
(c) By considering the values of $\tan\theta$ at the solutions found in part (a)(ii), and using part (b)(ii), show that $\tan 15^\circ = 2 - \sqrt3$ .

(2 marks)
Level 5 · 10 marksTry it
2
(a) Show that $\sin(A+B) - \sin(A-B) \equiv 2\cos A\sin B$ .
(b) By writing $3x = 2x + x$ and $x = 2x - x$ , solve $\sin 3x = \sin x$ for $0 \le x \le \pi$ , giving your answers as exact multiples of $\pi$ .
Level 5 · 6 marksTry it
3
The diagram shows triangle $PQR$ . The perpendicular from $R$ to $PQ$ meets $PQ$ at $N$ . Angle $PRN = x$ , angle $NRQ = y$ and the length $RN = h$ .
(a) (i) Find the length $PR$ in terms of $h$ and $x$ .

(ii) Write down the length $QR$ in terms of $h$ and $y$ .

(iii) By finding the area of triangle $PQR$ in two different ways, show that $\sin(x + y) \equiv \sin x\cos y + \cos x\sin y$ .

(6 marks)
(b) Given that $\sin(\alpha + 60^\circ) = \cos(\alpha - 45^\circ)$ , show that $\tan\alpha = \sqrt6 + \sqrt3 - \sqrt2 - 2$ .

(5 marks)
Level 5 · 11 marksTry it