A-level · Trigonometry

Graphs of sine, cosine and tangentExam-style questions

The shapes of y = sin x, cos x and tan x from the unit circle — period, amplitude, symmetry and asymptotes — and why reading them off the graph is how you find ALL the solutions to a trig equation.

Start practising Read the lesson

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

DifficultyAll Level 2 Level 3 Level 4

1
Figure 1

The curve $C_1$ has equation $y = 6\cos x$ , where $x$ is measured in radians. Figure 1 shows $C_1$ for $-2\pi \leqslant x \leqslant 2\pi$ .

On $C_1$ , the point $R$ is the first minimum point to the right of the $y$ -axis, and $S$ is a point where $C_1$ crosses the $x$ -axis, as shown in Figure 1.
(a) State

(i) the coordinates of $R$

(ii) the coordinates of $S$

(2 marks)

The curve $C_2$ has equation $y = 6\cos x + k$ , where $k$ is a constant. The point $\left(\dfrac{\pi}{3},\ 5\right)$ lies on $C_2$ .
(b) (i) Find the value of $k$ .

(ii) Write down the coordinates of the first maximum point on $C_2$ to the right of the $y$ -axis.

(3 marks)

The curve $C_3$ has equation $y = 6\cos(x - \alpha)$ , where $\alpha$ is a constant and $0 < \alpha < \pi$ . The curve $C_3$ crosses the $y$ -axis at the point $\left(0,\ 3\sqrt{3}\right)$ .
(c) (i) Find the value of $\alpha$ .

(ii) Write down the coordinates of the first minimum point on $C_3$ to the right of the $y$ -axis.

(2 marks)
(d) Use Figure 1 to find the number of solutions of the equation

$12\cos x = x \qquad -2\pi \leqslant x \leqslant 2\pi$

You must give a reason for your answer.

(2 marks)
Level 4 · 9 marksTry it
2
The diagram shows the graph of $y = \sin x°$ for $-180 \le x \le 540$ (a) The point $A\left(a, \dfrac{\sqrt{2}}{2}\right)$ is marked on the graph.
(a) (i) Find the value of $a$

(ii) State the value of $\sin(-a°)$

(3 marks)
(b) The point $B\left(b, -\dfrac{1}{3}\right)$ is also marked on the graph.
(b) (i) Find the exact value of $\cos(b° + 90°)$

(ii) Find the exact value of $\cos b°$

(5 marks)
Level 4 · 8 marksTry it