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

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

Solve the equation

$\text{tan}\left(3A-30°\right)=-\frac{1}{3}\sqrt{3}$

for $-120°\le A\le 120°$.

Solve the equation

$\frac{\text{sin}x}{\text{sec}x}=\frac{1}{4}$

for $-\pi \le x\le \pi$, giving your answers in terms of $\pi$.

Use the identity

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

to show that

$3\text{sin}\theta +4\text{cos}\theta$

can be expressed in the form

$5\text{sin}\left(\theta +\alpha \right)$

where

$\alpha ={\text{tan}}^{-1}\left(\frac{4}{3}\right)$

Hence solve the equation

$3\text{sin}\theta +4\text{cos}\theta =1$

for $0\le \theta \le \pi$, giving your answers correct to 3 significant figures.

State the greatest value of $3\text{sin}\theta +4\text{cos}\theta$ and find the least positive value of $\theta$ for which this greatest value occurs. Give your value of $\theta$ correct to 3 significant figures.

Show that the equation

$3{\text{tan}}^{2}x=18-2\text{sec}x$

can be expressed as

$3{\text{sec}}^{2}x+2\text{sec}x-21=0$

Hence solve the equation

$3{\text{tan}}^{2}x=18-2\text{sec}x$

for $-\pi \le x\le \pi$, giving your answers correct to 3 significant figures.

Solve the equation

$\text{cos}2\theta =\text{cos}\theta -1$

for $-\pi \le \theta \le \pi$, giving your answers in terms of $\pi$.

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

By drawing suitable horizontal lines on your sketch, show how the number of real roots of the equation

${\text{cot}}^2\theta =k$

for $-2\pi \le \theta \le 2\pi$, where $k$ is a constant, can be $0$, $4$ or $8$.

Solve the equation

${\text{cot}}^2\theta ={\text{sec}}^2\theta -1$

for $0°\le \theta \le 360°$.

Solve the equation ${\text{sec}}^{2}2x=1+\text{tan}2x$ for $0°\le x\le 180°$.

Given that

$\text{sin}\left(2A-B\right)=\frac{1}{4}\left(\sqrt{6}+\sqrt{2}\right)$

where $3A=4B$ and $60°<B<A<300°$, find the value of $A$ and the value of $B$.

Solve the equation

$\frac{\text{cos}x}{\text{cosec}x}-\text{cot}x=0$

for $-2\pi \le x\le 2\pi$, giving your answers in terms of $\pi$.

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

Hence solve the equation

$3\text{cos}\theta -4\text{sin}\theta -2=0$

for $0\le \theta \le 2\pi$, giving your answers correct to 3 significant figures.

State the least value of $6\text{cos}\theta -8\text{sin}\theta$ and find the least positive value of $\theta$ for which this least value occurs. Give your value of $\theta$ correct to 3 significant figures.

Solve the equation

$2{\text{cot}}^{2}x=8-\text{cosec}x$

for $-\pi \le x\le \pi$, giving your answers correct to 3 significant figures where appropriate.

Solve the equation

$8{\text{cos}}^4\theta -5\text{cos}2\theta -2=0$

for $0\le \theta \le \pi$, giving your answers in terms of $\pi$.

Find the set of values of the constant $k$ for which the equation

$\text{cosec}\theta =k$

for $-\pi \le \theta \le 2\pi$, has

(i) no real roots,

(ii) exactly 1 real root,

(iii) exactly 2 real roots,

(iv) exactly 4 real roots.

Solve the equation

${\text{cot}}^2\theta =15-6\text{cosec}\theta$

for $-180°\le \theta \le 180°$, giving your answers correct to 1 decimal place where appropriate.

Solve the equation

${\text{sin}}^{3}3\theta -\text{sin}3\theta {\text{cos}}^{2}3\theta =0$

for $0°\le \theta <180°$.

Solve the equation

$5\text{sin}\theta +2\text{cos}\theta =3$

for $-\pi \le \theta \le \pi$, giving your answers correct to 3 significant figures.

State the greatest value of $5\text{sin}\theta +2\text{cos}\theta$ and the second positive value of $\theta$ for which it occurs. Give your value of $\theta$ correct to 3 significant figures.

Solve the equation

$3{\text{sec}}^4\theta +16=16+16{\text{tan}}^2\theta$

for $-\pi \le \theta \le \pi$, giving your answers in terms of $\pi$.

Given that

$\text{cos}\left(A-B\right)=-\frac{\sqrt{3}}{2}$

and

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

where $0\le 2B<A\le 360°$, find the possible values of $A$ and $B$.

Solve the equation

${\text{cosec}}^{2}x-2\frac{\text{cosec}x}{\text{sec}x}=9$

for $0\le x\le 2\pi$, giving your answers correct to 3 significant figures.

Solve the equation

$8{\text{sin}}^{4}2\theta =2-5\text{cos}4\theta$

for $-\frac{1}{2}\pi \le \theta \le \frac{1}{2}\pi$, giving your answers in terms of $\pi$.

Given that $x=-2$ is a root of the equation

$x^3+12x^2+44x+48=0$

solve the equation by factorisation.

Hence solve the equation

$\text{sec}\theta \left({\text{sec}}^2\theta +44\right)+12\left({\text{tan}}^2\theta +5\right)=0$

for $0°\le \theta \le 180°$, giving your answers correct to 1 decimal place where appropriate.