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 shown in Figure 1 has equation $y = 7\sin x$ , where $x$ is measured in degrees.

On the curve, $A$ is a maximum point and $B$ is a point where the curve crosses the $x$ -axis, as shown in Figure 1.
(a) Write down the coordinates of $A$ .

(1 mark)
(b) Write down the coordinates of $B$ .

(1 mark)
(c) State the number of solutions of

(i) $7\sin x + 2 = 0$ for $-1800° < x < 1800°$

(ii) $7\sin x = 8$ for $0 \leqslant x \leqslant 3600°$

(iii) $3 - 7\sin x = 10$ for $0 < x \leqslant 2700°$

(3 marks)
Level 3 · 5 marksTry it
2
Figure 1

The curve with equation $y = \tan x$ , where $x$ is in radians, and the straight line $l$ with equation $2y = x$ are sketched in Figure 1 for $-\pi < x < \pi$ .
(a) State the equations of the two asymptotes to the curve $y = \tan x$ for $-\pi < x < \pi$ .

(2 marks)
(b) State how many roots the equation $2\tan x = x$ has for

(i) $-\pi < x < \pi$

(ii) $0 < x < 5\pi$

(iii) $-30\pi < x < 30\pi$

(3 marks)
Level 3 · 5 marksTry it
3
State the number of solutions of the equation $5\cos x^\circ = 2$ in the interval $0 \le x \le 900$ .
Level 3 · 2 marksTry it
4
On the same axes, sketch $y = 2\sin x^\circ$ and $y = 1$ for $0 \le x \le 360$ . State the number of points of intersection, and find the values of $x$ at which they occur.
Level 3 · 2 marksTry it
5
Describe fully the two transformations that map the graph of $y = \cos x^{\circ}$ onto the graph of $y = 4\cos\left(\dfrac{x}{2}\right)^{\circ}$ , and state the period of the new graph. (3 marks)
Level 3 · 3 marksTry it

7 more questions on graphs of sine, cosine and tangent

Free with an account — just an email, no card. The whole bank, every mark scheme and worked solution.

Start free