Radian Measure (AQA A Level Maths: Pure): Exam Questions

Given that $\theta$ is small and is measured in radians, use the small angle approximations to find an approximate value of

$\frac{1-\cos 4\theta }{2\theta \sin 3\theta }$

A parachutist, modelled by the point $O$ below, is heading towards the ground using a parachute, modelled by the shaded segment.

The radii $OA$ and $OB$ are strings that connect the parachutist to the parachute.

The area of sector $OAB$ is  $\frac{81\pi }{200}$m².

Find the length of one of the strings.

Given that $x$ is small and measured in radians, use small angle approximations to show that 

$16-7\sin x+2\tan 2x-18\cos x\approx a{x}^2+bx+c$

where $a$, $b$ and $c$ are constants to be found.

Hence find two approximate solutions to the equation

$16-7\sin x+2\tan 2x-18\cos x=0$

Find all the solutions to the equation

$\cos \theta =\frac{1}{2}$

in the interval  $-2\pi \le \theta \le 2\pi$, giving your answers in exact form.

Find all the solutions to the equation

$5\sin 3x=1$

in the interval  $0\le x\le \pi$, giving your answers to 3 significant figures.

The sector $OAB$ is shown in the diagram below, where

• the central angle of the sector is $\frac{\pi }{3}$ radians

• the line segments $OC$ and $BC$ have lengths of $8$ cm and $p$cm respectively

• $CD$ is parallel to $AB$

Find an expression for the exact area of sector $OAB$ in terms of $p$.

Find the exact area of triangle $OCD$.

The shaded shape $ABCD$, where $AB$ is the arc of the sector, has an area of

$\left(\frac{50\pi }{3}-16\sqrt{3}\right)$ cm²

Find the value of $p$.

The diagram below shows the sector $OAB$.

Find the area of the shaded segment, giving your answer to 3 significant figures.

Find the perimeter of the sector $OAB$, to 2 decimal places.

The diagram shows a prism where

• the uniform cross section of the prism is a sector

• the radius of the sector is $\left(2a-3\right)$ cm

• the central angle of the sector is $1.7$ radians

• the height of the prism is $4$ cm

Given that the volume of the prism is $7.65$ cm³, find the value of $a$.

Given that $\theta$ is small and measured in radians, use the small angle approximations to find an approximate value of

$\frac{{\sin }^2\theta +\cos \theta }{\tan \theta +\sin \theta }$

A sector $OPQ$ of a circle centre $O$ has a radius of $a$ cm and a central angle of $a^2\pi$ radians, where $a$ is a non-zero constant.

The area of sector $OPQ$ is three times the length of the arc $PQ$.

Find the value of $a$.

Given that $\theta$ is small and measured in radians, use the small angle approximations to find and simplify an approximate value of

$\frac{{\sin }^2\theta +2{\cos }^2\left(2\theta \right)-\tan \left(3\theta \right)}{\sin \left(4\theta \right)}$

Given that $x$ is small and measured in radians, use small angle approximations to show that 

${\cos }^2\left(3x\right)+\frac{1}{4}{\tan }^4\left(3x\right)$

can be approximated by

$1+a{x}^2+b{x}^4$

where $a$ and $b$ are constants to be found.

Hence find three approximate solutions to the equation

${\cos }^2\left(3x\right)+\frac{1}{4}{\tan }^4\left(3x\right)=1$

Comment on the validity of your solutions in part (b).

The sector $OAB$ is shown in the diagram below, where

• the central angle of the sector is $\theta$ radians

• the radii $OA$ and $OB$ are both equal to $r$

• $AD=BC=1$ cm

Find an expression for the shaded area $ABCD$ in terms of $r$ and $\theta$.

Show that, for values of $\theta$ near to zero, an approximation of the shaded area $ABCD$ is given by the formula

$\frac{1}{2}\theta \left(ar+b\right)$

where $a$ and $b$ are constants to be found.

The diagram below shows the sector $OAB$.

Show that the area of the shaded segment is

$\frac{1}{2}{r}^2\left(\theta -\sin \theta \right)$ cm²

Find, in terms of $\theta$, an expression for the percentage of the sector that the segment occupies.

Given that $\theta$ is small and measured in radians, use the small angle approximations to find approximate solutions to the equation

$\frac{4{\cos }^2\left(4\theta \right)-{\sin }^2\left(2\theta \right)}{{\tan }^2\theta }=12$

Determine which out of the solutions in part (a) are more accurate, giving a reason for your answer.

A sector $OST$ of a circle centre $O$ has a radius of $r$ cm and a central angle of $\theta$ radians.

The chord $ST$ has length $a$ cm.

Show that 

$a^2=2r^2\left(1-\cos \theta \right)$

Given that

• $r=4\theta$

• the area of the sector $OST$ is $\frac{8{\pi }^3}{27}$ cm²

find the exact value of $a$.

The sector $OAB$ is shown in the diagram below, where

• the central angle of the sector is $\theta$ radians

• the radii $OA$ and $OB$ are both equal to $r$

• The line $DC$ is perpendicular to the line $OB$

• $BC:CO=2:3$

Show that the area of the shaded shape $ABCD$ is

$\frac{1}{50}{r}^2\left(a\mathrm{\theta }+b\tan \theta \right)$ cm² 

where $a$ and $b$ are constants to be found.