AS-Level Maths | Trigonometry

Key Reference

Corresponding values

$\sin^{-1}(x)$ — use $180 - \text{ans}$ for the second solution

$\cos^{-1}(x)$ — use $360 - \text{ans}$ for the second solution

$\tan^{-1}(x)$ — use $\text{ans} + 180$ for the second solution

For all: after finding two consecutive values, keep adding or subtracting 360° to find more values if needed.

Question 1

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\cos x° = 0.185$

(b) $\cos x° = 0.4$

(c) $\cos x° = -0.39$

(d) $\cos x° = -0.636$

Question 2

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\sin x° = 0.27$

(b) $\sin x° = 0.813$

(c) $\sin x° = -0.203$

(d) $\sin x° = -0.6$

Question 3

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\tan x° = 0.1$

(b) $\tan x° = 1.6$

(c) $\tan x° = -0.7$

(d) $\tan x° = -3.4$

Question 4

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\cos(x + 45)° = 0.9$

(b) $\cos(x + 45)° = -0.5$

(c) $\cos(x - 45)° = 0.2$

(d) $\cos(x - 60)° = 0.6$

(e) $\cos 2x° = 0.64$

(f) $\cos 2x° = -0.415$

Question 5

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\sin(x - 60)° = 0.5$

(b) $\sin(x + 30)° = 0.14$

(c) $\sin(x - 60)° = -0.89$

(d) $\sin(x - 30)° = -0.3$

(e) $\sin 2x° = 0.5$

(f) $\sin 2x° = -0.18$

(g) $\sin \tfrac{1}{2}x° = 0.703$

Question 6

Solve each equation for $x$ in the interval $0 \leq x \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\tan(x + 30)° = 1$

(b) $\tan(x - 60)° = -1.26$

(c) $\tan 2x° = -2.74$

(d) $\tan 3x° = 0.591$

Trigonometric Identities

$\sin^2(x) + \cos^2(x) = 1$

$\dfrac{\sin(x)}{\cos(x)} = \tan(x)$

Question 7

Solve each equation for $\theta$ in the interval $0 \leq \theta \leq 360°$. Give your answers to 1 decimal place where appropriate.

(a) $\sin^2\theta° = 0.75$

(b) $1 - \tan^2\theta° = 0$

(c) $2\cos^2\theta° + \cos\theta° = 0$

(d) $\sin\theta°(4\cos\theta° - 1) = 0$

(e) $4\sin\theta° = \sin\theta°\tan\theta°$

(f) $(2\cos\theta° - 1)(\cos\theta° + 1) = 0$

(g) $\tan^2\theta° - 3\tan\theta° + 2 = 0$

(h) $3\sin^2\theta° - 7\sin\theta° + 2 = 0$

Question 8

Solve each of the following trigonometric equations in the range given.

(a) $2\cos^2 x - \sin x = 1$    $0 \leq x < 360°$

(b) $2\cos^2 x + 1 = 5\sin x$    $0 \leq x < 360°$

(c) $2\cos^2 x + 7\sin x + 2 = 0$    $0 \leq x < 360°$

Question 9

Solve each of the following trigonometric equations in the range given.

(a) $2\sin^2 x - 5\cos x + 1 = 0$    $0 \leq x < 360°$

(b) $5\sin^2 y = 19\cos y + 1$    $0 \leq y < 360°$

(c) $3\sin^2\theta = 8 + 16\cos\theta$    $0 \leq \theta < 360°$

Question 10

Solve each of the following trigonometric equations in the range given.

(a) $5\sin x + 6\cos x = 0$    $0 \leq x < 360°$

(b) $3\sin 2\theta = \sqrt{3}\cos 2\theta$    $-180 \leq \theta < 180°$

(c) $\sin 3\theta = \cos 3\theta$    $0 \leq \theta < 180°$

(d) $\cos x = 3\sin x$    $0 \leq x < 360°$

(e) $4\cos x - 7\sin x = 0$    $0 \leq x < 360°$

(f) $5\cos 2x + 9\sin 2x = 0$    $0 \leq x < 180°$

Question 11

Solve each of the following trigonometric equations in the range given.

(a) $3\cos x = 8\tan x$    $0 \leq x < 360°$

(b) $2\tan x\sin x = 3$    $0 \leq x < 360°$

(c) $2\cos x = 3\tan x$    $0 \leq x < 360°$

Question 12

Solve each of the following trigonometric equations in the range given.

(a) $4\tan^2\theta\cos\theta = 15$    $0 \leq \theta < 360°$

(b) $\dfrac{3 + \sin^2\theta}{\cos\theta - 2} = 3\cos\theta$    $0 \leq \theta < 360°$

(c) $2\sin x = \tan x$    $0 \leq x < 360°$

Question 13

Solve each of the following trigonometric equations in the range given.

(a) $\tfrac{1}{2}\tan x - \sin x = 0$    $0 \leq x < 360°$

(b) $\dfrac{\sin x - \cos x}{\cos x} = 2$    $0 \leq x < 360°$

(c) $4\tan y\sin y\cos y + 4\tan y\cos y + 1 = 0$    $0 \leq y < 360°$