Further Trigonometric Equations (AQA A Level Maths: Pure): Exam Questions

Solve the equation

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

for $-\pi \le x\le \pi$.

Give your answers to 3 significant figures, where necessary.

State the domain and range of the function

$\mathrm{f}\left(x\right)=\arcsin x$

Solve the equation

${\left[\mathrm{f}\left(x\right)\right]}^2={\displaystyle \frac{{\pi }^2}{16}}$

Use a small angle approximation to estimate the solution to the equation

$\mathrm{cosec}\theta =10$

Find the exact solution to

$\mathrm{cosec}\theta =10$

for $0<\theta <\frac{\mathrm{\pi }}{2}$ in the form

$\arcsin \left(\frac{1}{k}\right)$

where $k$ is an integer to be found.

Find, to 2 significant figures, the percentage error in using the approximate solution from part (a) instead of the the exact solution from part (b).

Solve the equation

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

for $-180°\le \theta \le 180°$.

Give your answers to one decimal place, where necessary.

Solve the equation

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

for $-\pi \le x\mathit{\le }\pi$.

Given that

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

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

Show that the equation 

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

can be written in the form

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

Hence solve the equation

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

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

Solve the equation

$\cos 2\theta =\cos \theta -1$

for $-\pi \le \theta \le \pi$, giving your answers in an exact form.

Write

$3\sin \theta +4\cos \theta$

in the form

$R\sin \left(\theta +\mathrm{\alpha }\right)$

where $R>0$ and $0<\alpha <\frac{\pi }{2}$.

Give $\alpha$ in the form ${\tan }^{-1}\left(\frac{p}{q}\right)$ where $p$ and $q$ are integers to be found.

Hence solve the equation

$3\sin \theta +4\cos \theta =1$

for  $0\le \theta \le \pi$.

Give your answers to 3 significant figures.

State the maximum value of

$3\sin \theta +4\cos \theta$

and find the smallest positive value of $\theta$ for which this maximum value occurs,  to 3 significant figures.

Solve the equation

${\cot }^2\theta -\cos \theta {\mathrm{cosec}}^2\theta =0$

for $0<\theta <2\pi$.

Solve the equation

${\sec }^2\left(2x\right)=1+\tan \left(2x\right)$

for  $0°\le x\le 180°$.

Prove the identity

$\frac{1}{\sin 2\theta }-\cot 2\theta \equiv \tan \theta ,\theta \ne \frac{n\pi }{2}$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Hence solve, for $0\le x<2\pi$, the equation

$\mathrm{cosec}4x-\cot 4x=\sqrt{3}$

Given that

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

and that

$3A=4B$ 

where $60°<B<A<300°$, find the values of $A$ and $B$.

Solve the equation

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

for $-2\pi \le x\le 2\pi$.

Write

$6\cos \theta -8\sin \theta$

in the form

$R\cos \left(\theta +\mathrm{\alpha }\right)$

where $R>0$ and $0<\alpha <\frac{\pi }{2}$, giving $\alpha$ to 3 significant figures.

Solve the equation

$3\cos \theta -4\sin \theta -2=0$

for  $0\le \theta \le 2\pi$.

Give your answers to 3 significant figures.

State the minimum value of

$6\cos \theta -8\sin \theta$

and find the smallest positive value of $\theta$ for which this minimum value occurs,  to 3 significant figures.

Solve the equation

$8{\cos }^4\theta -5\cos \left(2\theta \right)-2=0$

for $0\le \theta \le \pi$, giving your answers in an exact form.

On the same axes, sketch the curves $y=\arcsin x$ and  $y=\arccos x$, where $x$ is in radians.

State the solution to the equation

$\arccos x=\arcsin x$

Use a small angle approximation to estimate the positive solution to the equation

$\sec 2\theta =1.05$

Give your answer to 6 decimal places.

Find the exact solution to

$\sec 2\theta =1.05$

for $0<\theta <\frac{\pi }{2}$ in the form

$\frac{1}{p}\arccos \left(\frac{q}{r}\right)$

where $p$, $q$ and $r$ are integers to be found.

Find, to 2 significant figures, the percentage error in using the approximate solution from part (a) instead of the the exact solution from part (b).

Solve the equation

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

for $0°\le \theta \le 360°$.

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

$\mathrm{cosec}\theta =k$

for $-\pi \le \theta \le 2\pi$ has

(i) no real solutions

(ii) 1 real solution

(iii) 2 real solutions

(iv) 4 real solutions

Solve the equation

${\sin }^3\left(3\theta \right)-\sin \left(3\theta \right){\cos }^2\left(3\theta \right)=0$

for  $0°\le \theta <180°$.

Given that

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

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

• $0\le 2B<A\le 360°$

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

Sketch the curve $y=\pi +\arctan \left(2x-1\right)$ for $x\in \mathrm{ℝ}$.

Label clearly the coordinates of the point at which the curve meets the $y$-axis and state the equations of any asymptotes.

A different curve is given by $y=\pi +\arctan \left(1-2x\right)$ for $x\in \mathrm{ℝ}$.

Find the coordinates of the point of intersection of this curve with

(i) the $y$-axis,

(ii) the curve in part (a).

Solve the equation

$5\sin \theta +2\cos \theta =3$

for $-\pi \le \theta \le \pi$.

Give your answers to 3 significant figures.

State the maximum value of

$5\sin \theta +2\cos \theta$

and find the second smallest positive value of $\theta$ for which this maximum value occurs,  to 3 significant figures.

Solve the equation

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

for $0\le x\le 2\pi$.

Give your answers to 3 significant figures.

Find the domain and range of the function

$\mathrm{f}\left(x\right)=\cos \left(\arcsin x\right)$

Solve the equation

$2{\left[\mathrm{f}\left(x\right)\right]}^2-3\mathrm{f}\left(x\right)+1=0$

Find the value of

(i) $\arcsin \left(\sin \frac{7\pi }{6}\right)$

(ii) $\arccos \left(\cos \left(-\frac{2\pi }{3}\right)\right)$

Explain why

$\arctan \left(\tan \pi \right)\ne \pi$

Solve the equation

$8{\sin }^4\left(2\theta \right)=2-5\cos \left(4\theta \right)$

for $-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$.

Give your answers in an exact form.

The number of real solutions, $n$, to the equation

$\left|\sec x-2\right|=k$

where $-2\pi \le x\le 2\pi$ is determined by the value of the constant $k$, where $k\in \mathrm{ℝ}$.

Find the possible values of $n$ and state the corresponding range of values of $k$ for each $n$.

Given that $x=-2$ is a solution to the cubic equation

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

solve the equation by factorisation.

Solve

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

where $0°\le \theta \le 180°$.

Give your answers to 1 decimal place, where necessary.