Further Trigonometric Equations (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Solve the equation cosec $2\theta =2$ for $0°\le \theta \le 180°$.

Given that

tan$\left(3A°-30°\right)=-\frac{\sqrt{3}}{3}$

find the values of $A$ such that  $-120°\le A°\le 120°$.

Solve the equation

$\frac{\sin x}{\sec x}=\frac{1}{4},-\mathrm{\pi }\le x\mathit{\le }\mathrm{\pi }$

Use the identity

$R$sin$\left(A+B\right)\equiv R$cos $B$ sin $A+R$sin $B$ cos $A$

to show that

$3$ sin $\theta +4$ cos $\theta$

can be written as

$5$sin$\left(\theta +\mathrm{\alpha }\right),\mathrm{where}\alpha ={\tan }^{-1}\left(\frac{4}{3}\right)$

Hence, or otherwise, solve the equation $3$ sin $\theta +4$ cos $\theta =1$ for  $0\le \theta \le \mathrm{\pi }$.
Give your answers to three significant figures.

Write down the maximum value of $3$ sin $\theta +4$cos $\theta$ and state the first positive value of $\theta$ for which it occurs.  Give your value of $\theta$  to three significant figures.

Show that the equation  $3$ tan² $x=18-2$ sec $x$  can be written as

$3$ sec² $x+2$sec $x-21=0$

Hence, or otherwise, solve the equation

$3$ tan² $x=18-2$ sec $x,-\mathrm{\pi }\le x\le \mathrm{\pi }$

Give your answers to three significant figures.

Solve the equation

cos $2\theta =$cos $\theta -1-\mathrm{\pi }\le \theta \le \mathrm{\pi }$

State your answers as multiples of $\mathrm{\pi }$.

Sketch the graph of $y={\cot }^2\theta$ for  $-2\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$.

By adding three lines to your graph demonstrate how the equation

cot² $\theta =\mathrm{k}-2\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$

where $k$ is a constant has either 0, 4 or 8 real solutions.

Solve the equation

cot² $\theta =$sec² $\theta -1,0°\le \theta \le 360°$

Solve the equation  sec² $2x=1+$tan $2x$  for  $0°\le x\le 180°$.

Given that

sin$\left(2A°-B°\right)=\frac{\sqrt{6}+\sqrt{2}}{4}$

and that

$3A=4B$  and $60°<B°<A°<300°$

find the values of $A$ and $B$.

Solve the equation

$\frac{\cos x}{\mathrm{cosec}x}-\cot x=0,-2\mathrm{\pi }\le x\le 2\mathrm{\pi }$

Show that $6$ cos $\theta -8$ sin $\theta$ can be written in the form  $R$ cos$\left(\theta +\mathrm{\alpha }\right)$, where $R>0$ and $\alpha$ is an acute angle measured in radians.

Hence, or otherwise, solve the equation $3$ cos $\theta -4$ sin $\theta -2=0$,for $0\le x\le 2\mathrm{\pi }$
Give your answers to three significant figures.

Write down the minimum value of $6$ cos $\theta -8$ sin $\theta$ and the smallest positive value of $\theta$ for which it occurs.  Give your value of $\theta$  to three significant figures.

Solve the equation

$2$ cot² $x=8-$cosec $x,-\mathrm{\pi }\le x\le \mathrm{\pi }$

giving your answers to three significant figures where appropriate.

Solve the equation

$8$cos⁴ $\theta -5$ cos $2\theta -2=00\le \theta \le \mathrm{\pi }$

State your answers as multiples of $\mathrm{\pi }$.

Determine the values of the constant $k$ for which the equation

                  cosec $\theta =\mathrm{k},-\mathrm{\pi }\le \theta \le 2\mathrm{\pi }$

has      (i) no real solutions,
            (ii) 1 real solution,
            (iii) 2 real solutions,
            (iv) 4 real solutions

Solve the equation

cot² $\theta =15-6$cosec $\theta ,-180°\le \theta \le 180°$

Give your answers to one decimal place where appropriate.

Solve the equation sin³ $3\theta -$sin $3\theta$ cos² $3\theta =0$  for  $0°\le \theta <180°$.

Given that

cos$\left(A°-B°\right)=-\frac{\sqrt{3}}{2}$ and  tan$\left(\frac{1}{2}A°-B°\right)=\sqrt{3}$

and that

$0\le 2B°<A°\le 360°$

find the possible values of $A$ and $B$.

Solve the equation $5$ sin $\theta +2$ cos $\theta =3$,for  $-\mathrm{\pi }\le \theta \le \mathrm{\pi }$.
Give your answers to three significant figures.

Write down the maximum value of $5$ sin $\theta +2$ cos $\theta$ and the second positive value of $\theta$  for which it occurs.  Give your value of  to three significant figures.

Solve the equation

$3$ sec⁴ $\theta +16=16+16$tan² $\theta ,-\mathrm{\pi }\le \theta \le \mathrm{\pi }$

giving your answers to three significant figures where appropriate.

Solve the equation

cosec² $x-2\frac{\mathrm{cosec}x}{\sec x}=90\le x\le 2\mathrm{\pi }$

Give your answers to three significant figures.

Solve the equation

$8$sin⁴ $2\theta =2-5$ cos $4\theta -\frac{\mathrm{\pi }}{2}\le \theta \le \frac{\mathrm{\pi }}{2}$

State your answers as multiples of $\mathrm{\pi }$.

The number of real solutions to the equation

$\left|\sec x-2\right|=k,-2\mathrm{\pi }\le x\le 2\mathrm{\pi }$

is determined by the value of the constant $k$.

Find the number of real solutions for all values of $k$, given that $k\in \mathrm{ℝ}$.

Given that $x=-2$ is a root of $x^3+12x^2+44x+48$, solve the equation by factorisation.

Solve the equation

$\sec \theta \left({\sec }^2\theta +44\right)+12\left({\tan }^2\theta +5\right)=0,0°\le \theta \le 180°$

Give your answers to one decimal place where appropriate.