A-level · Trigonometry

The R-form (a sinθ + b cosθ)Exam-style questions

Writing a sinθ ± b cosθ as a single R sin(θ ± α) or R cos(θ ∓ α), finding R and α by matching coefficients, and using the form to read off maxima/minima, solve equations and answer modelling questions.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
(a) Express $6\cos 2x + 5\sin 2x$ in the form $R\cos(2x - \alpha)$ , where $R > 0$ and $0 < \alpha < \tfrac12\pi$ . Give the exact value of $R$ and give the value of $\alpha$ correct to 4 significant figures.

(3 marks)
(b) Give full details of a sequence of three transformations that maps the curve $y = \cos x$ onto the curve $y = 6\cos 2x + 5\sin 2x$ .

(4 marks)
(c) Find the maximum value of $6\cos 2x + 5\sin 2x$ , and determine the smallest positive value of $x$ at which this maximum is reached.

(2 marks)
(d) Find the minimum value of $6\cos 2x + 5\sin 2x$ , and determine the smallest positive value of $x$ at which this minimum is reached.

(2 marks)
Level 5 · 11 marksTry it
2
(a) Express $7\cos\theta + 24\sin\theta$ in the form $R\cos(\theta - \alpha)$ , with $R > 0$ and $0 < \alpha < 90^\circ$ .
(b) State the maximum value and the value of $\theta$ (to 1 d.p.) at which it occurs.
(c) Solve $7\cos\theta + 24\sin\theta = 15$ for $0 \le \theta \le 360^\circ$ , to 1 d.p.
Level 5 · 8 marksTry it
3
(a) Express $5\sin\theta - 12\cos\theta$ in the form $R\sin(\theta - \alpha)$ , where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$ . Give $\alpha$ correct to 4 significant figures.
(b) Hence solve $5\sin\theta - 12\cos\theta = 6.5$ for $0 \le \theta \le 2\pi$ , giving your answers correct to 3 significant figures.
Level 5 · 7 marksTry it
4
(a) Express $2\cos x - 2\sqrt{3}\sin x$ in the form $R\cos(x + \alpha)$ , where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$ . (3 marks)

The depth of water, $h$ metres, in a harbour $t$ hours after midnight is modelled by

$h = 5 + 2\cos\left(\dfrac{\pi t}{6}\right) - 2\sqrt{3}\sin\left(\dfrac{\pi t}{6}\right), \qquad 0 \le t \le 24.$

A boat can enter the harbour only when the depth of water is at least $7$ metres.
(b) Find the total length of time, during these $24$ hours, for which the boat is able to enter the harbour. Fully justify your answer. (6 marks)
Level 5 · 9 marksTry it
5
The depth of water, $D$ metres, at the entrance to a harbour is modelled by

$D = 9 + 3\sin\left(\dfrac{\pi t}{6}\right) - 3\sqrt{3}\cos\left(\dfrac{\pi t}{6}\right)$ , $\qquad 0 \le t < 24$ ,

where $t$ is the time in hours after midnight.
(a) Express $3\sin\theta - 3\sqrt{3}\cos\theta$ in the form $R\sin(\theta - \alpha)$ , where $R > 0$ and $0 < \alpha < \dfrac{\pi}{2}$ , stating the exact values of $R$ and $\alpha$ . (3 marks)
(b) Find the maximum depth of water at the harbour entrance, and the first time after midnight at which this maximum depth occurs. (2 marks)

A ship can enter the harbour only when the depth of water is at least $12$ metres.
(c) Find the times during the day between which the ship can enter the harbour. (4 marks)
Level 5 · 9 marksTry it

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