A-Level Maths | Trigonometry

Question 1

Solve each of the following trigonometric equations.

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

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

c) $2\cosec 2y = 10, \quad 0 \leq y < 2\pi$

d) $8\tan\varphi = \cot^2\varphi, \quad 0 \leq \varphi < 2\pi$

e) $2\sec\theta = 3, \quad 0 \leq \theta < 360°$

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

g) $5 - \cosec 2y = -1, \quad 0 \leq y < 2\pi$

h) $27\sin^2\varphi + 8\cosec\varphi = 0, \quad 0 \leq \varphi < 2\pi$

Question 2

Solve each of the following equations.

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

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

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

d) $2\cosec^2\varphi + \cot^2\varphi = 11, \quad 0 \leq \varphi < 360°$

Question 3

Prove the validity of each of the following trigonometric identities.

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

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

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

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

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

Question 4

Solve each of the following trigonometric equations.

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

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

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

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

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

Question 5

Prove the validity of each of the following trigonometric identities.

a) $\sec\theta\cosec\theta \equiv 2\cosec 2\theta$

b) $\tan\theta + \cot\theta \equiv 2\cosec 2\theta$

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

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

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

f) $\cot 2x + \cosec 2x \equiv \cot x$

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

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

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

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

Question 6

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

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

b) State the minimum and maximum value of:

 i. $y = f\!\left(x - \dfrac{\pi}{2}\right)$

 ii. $y = 2f(x) + 1$

 iii. $y = \left[f(x)\right]^2$

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

Question 7

$y \equiv 2\sqrt{2}\cos x + 2\sqrt{2}\sin x,\ x\in\mathbb{R}$

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

b) Solve the equation $y = 2$ for $0 < x < 2\pi$.

c) Write down the maximum value of $y$.

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

Question 8

a) Express $3\cos 3x + 7\sin 3x$ in the form $R\cos(3x - \alpha)$, where $R > 0$ and $0 < \alpha < \dfrac{1}{2}\pi$.

b) Give full details of a sequence of three transformations needed to transform the curve $y = \cos x$ to the curve $y = 3\cos 3x + 7\sin 3x$.

c) Determine the greatest value of $3\cos 3x + 7\sin 3x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs.

d) Determine the least value of $3\cos 3x + 7\sin 3x$ as $x$ varies and give the smallest positive value of $x$ for which it occurs.

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