A-Level Maths | Trigonometry

Question 1

Solve each of the following trigonometric equations.

a) $\sec x = 4, \quad 0 \leq x < 360°$

b) $3\cot 2x - 1 = 4, \quad 0 \leq x < 180°$

c) $2\sec x = 3, \quad 0 \leq x < 360°$

d) $\cot 3x = \dfrac{1}{4}, \quad -90° \leq x < 90°$

e) $2\operatorname{cosec} 2x = 10, \quad 0 \leq x < 2\pi$

f) $8\tan x = \cot^2 x, \quad 0 \leq x < 2\pi$

g) $5 - \operatorname{cosec} 2x = -1, \quad 0 \leq x < 2\pi$

h) $27\sin^2 x + 8\operatorname{cosec} x = 0, \quad 0 \leq x < 2\pi$

Question 2

Solve each of the following equations.

a) $2\tan^2 x = 11\sec x - 7, \quad 0 \leq x < 360°$

b) $4\cot^2 x - 9\operatorname{cosec} x + 6 = 0, \quad 0 \leq x < 360°$

c) $\sec^2 x + \tan x = 3, \quad 0 \leq x < 360°$

d) $2\operatorname{cosec}^2 x + \cot^2 x = 11, \quad 0 \leq x < 360°$

Question 3

Prove the validity of each of the following trigonometric identities.

a) $\sec x - \sec x\sin^2 x \equiv \cos x$

b) $(\cos x + \sec x)^2 \equiv \tan^2 x + \cos^2 x + 3$

c) $\cos x\sin x(\cot x + \tan x) \equiv 1$

d) $\cos x + \sin x\tan x \equiv \sec x$

e) $\dfrac{\cot^2 x}{1 + \cot^2 x} \equiv \cos^2 x$

f) $\dfrac{1}{\sec x - \tan x} + \dfrac{1}{\sec x + \tan x} \equiv 2\sec x$

g) $\dfrac{\tan x\sec x}{1 + \tan^2 x} \equiv \sin x$

h) $(1 - \cos x)(1 + \sec x) \equiv \sin x\tan x$

i) $\dfrac{\operatorname{cosec} x}{\operatorname{cosec} x - \sin x} \equiv \sec^2 x$

j) $(1 - \sin x)(1 + \operatorname{cosec} x) \equiv \cos x\cot x$

k) $\dfrac{\cot x\operatorname{cosec} x}{1 + \cot^2 x} \equiv \cos x$

l) $(\operatorname{cosec} x - \sin x)\sec^2 x \equiv \operatorname{cosec} x$

m) $\dfrac{1}{1 + \sin^2 x} + \dfrac{1}{1 + \operatorname{cosec}^2 x} \equiv 1$

n) $\dfrac{\sin x}{1 + \cos x} + \dfrac{1 + \cos x}{\sin x} \equiv 2\operatorname{cosec} x$

o) $\left(\dfrac{1 + \sin x}{\cos x}\right)^2 + \left(\dfrac{1 - \sin x}{\cos x}\right)^2 \equiv 2 + 4\tan^2 x$

Question 4

Solve each of the following trigonometric equations.

a) $\cos(x + 30°) = \sin x, \quad 0 \leq x < 360°$

b) $3\cos(x + 30°) = \sin(x - 60°), \quad 0 \leq x < 360°$

c) $\sin(x - 30°) = \sin(x + 45°), \quad 0 \leq x < 360°$

d) $\sin(x + 30°) = \cos(x - 45°), \quad 0 \leq x < 360°$

e) $\cos(x - 60°) = \cos(x - 45°), \quad 0 \leq x < 360°$

Question 5

Prove the validity of each of the following trigonometric identities.

a) $\sec x\operatorname{cosec} x \equiv 2\operatorname{cosec} 2x$

b) $\tan x + \cot x \equiv 2\operatorname{cosec} 2x$

c) $\dfrac{1 - \cos 2x}{\sin 2x} \equiv \tan x$

d) $\dfrac{\cos 2x}{\cos x - \sin x} \equiv \cos x + \sin x$

e) $\dfrac{\cos 2x}{\sin x} + \dfrac{\sin 2x}{\cos x} \equiv \operatorname{cosec} x$

f) $\cot 2x + \operatorname{cosec} 2x \equiv \cot x$

g) $\cos 2x + \tan x\sin 2x \equiv 1$

h) $\dfrac{\sin x}{1 - \cos x} \equiv \cot\dfrac{x}{2}$

i) $\sin 2x \equiv \dfrac{2\tan x}{1 + \tan^2 x}$

j) $\dfrac{1}{\cos x - \sin x} - \dfrac{1}{\cos x + \sin x} \equiv 2\sin x\sec 2x$

Question 6

$f(x) \equiv 9\sin x + 12\cos x,\ x\in\mathbb{R}$

a) Express $f(x)$ in the form $R\sin(x + \alpha)$, $R > 0$, $0 < \alpha < 90°$.

$f(x) \equiv 15\sin(x + 53.1°)$

b) Hence write down the minimum and maximum values of $f(x)$.

Question 7

$f(x) \equiv 4\sin x + 3\cos x,\ x\in\mathbb{R}$

a) Express $f(x)$ in the form $R\sin(x + \alpha)$, $R > 0$, $0 < \alpha < 90°$.

$f(x) \equiv 5\sin(x + 36.9°)$

b) Hence write down the maximum value of $f(x)$.

c) Find the smallest positive value of $x$ for which this maximum value occurs.

Question 8

$f(x) \equiv \sin x - \sqrt{3}\cos x,\ x\in\mathbb{R}$

a) Express $f(x)$ in the form $R\sin(x - \alpha)$, $R > 0$, $0 < \alpha < \dfrac{\pi}{2}$.

$f(x) \equiv 2\sin\left(x - \dfrac{\pi}{3}\right)$

b) Hence write down the maximum value of $f(x)$.

c) Find the smallest positive value of $x$ for which this maximum value occurs.

Question 9

$f(x) \equiv 5\sin x - 2\cos x,\ 0 < x < 2\pi$

a) Express $f(x)$ in the form $R\sin(x - \alpha)$, $R > 0$, $0 < \alpha < \dfrac{\pi}{2}$.

$f(x) \equiv \sqrt{29}\sin(x - 0.381)$

b) Hence write down the minimum value of $f(x)$.

c) Find the smallest positive value of $x$ for which this minimum value occurs.

Question 10

$f(x) \equiv 3\sin x + \cos x,\ x\in\mathbb{R}$

a) Express $f(x)$ in the form $R\cos(x - \alpha)$, $R > 0$, $0 < \alpha < \dfrac{\pi}{2}$.

$f(x) \equiv \sqrt{10}\cos(x - 1.249)$

b) Hence write down the minimum value of $f(x)$.

c) Find the smallest positive value of $x$ for which this minimum value occurs.

Question 11

$f(x) \equiv \sqrt{2}\cos x - \sqrt{6}\sin x,\ 0 < x < 360°$

a) Express $f(x)$ in the form $R\cos(x + \alpha)$, $R > 0$, $0 < \alpha < 90°$.

$f(x) \equiv \sqrt{8}\cos(x + 60°)$

b) Hence write down the minimum value of each of the following.

i) $[f(x)]^2$

ii) $\dfrac{1}{[f(x)]^2}$

Question 12

$f(x) \equiv 2\sin x + 2\cos x,\ x\in\mathbb{R}$

a) Express $f(x)$ in the form $R\sin(x + \alpha)$, $R > 0$, $0 < \alpha < \dfrac{\pi}{2}$.

$f(x) \equiv \sqrt{8}\sin\left(x + \dfrac{\pi}{4}\right)$

b) Hence state the minimum and maximum values of each of the following.

i) $f\left(x - \dfrac{\pi}{2}\right)$

ii) $2f(x) + 1$

iii) $[f(x)]^2$

iv) $\dfrac{10}{f(x) + 3\sqrt{2}}$