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

Figure 1 shows a sector $AOB$ of a circle with centre $O$ and radius $r$ cm.

The angle $AOB$ is $\theta$ radians.

The area of the sector $AOB$ is 11 cm².

Given that the perimeter of the sector is 4 times the length of the arc $AB$, find the exact value of $r$.

The shape $OABCDEFO$ shown in Figure 1 is a design for a logo.

In the design

• $OAB$ is a sector of a circle centre and radius $r$

• sector $OFE$ is congruent to sector $OAB$

• $ODC$ is a sector of a circle centre $O$ and radius $2r$

• $AOF$ is a straight line

Given that the size of angle $COD$ is $\theta$ radians, write down, in terms of $\theta$, the size of angle $AOB$

Show that the area of the logo is

$\frac{1}{2}{r}^2\left(3\theta +\pi \right)$

Find the perimeter of the logo, giving your answer in simplest form in terms of $r$, $\theta$ and $\pi$.

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

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

Given that $\theta$ is small and measured in radians, use the small angle approximations to show that

$4\sin \frac{\theta }{2}+3{\cos }^2\theta \approx a+b\theta +c{\theta }^2$

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

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.

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 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

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

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

• face $ABC$ is a sector of a circle with radius $r$ cm and centre $A$

• angle $BAC=0.8$ radians

• faces $ABC$ and $DEF$ are congruent

• edges $AD,CF$ and $BE$ are perpendicular to faces $ABC$ and $DEF$

• edges $AD,CF$ and $BE$ have length $h$ cm

Given that the volume of the toy is $240{\mathrm{cm}}^3$ show that the surface area of the toy, $S{\mathrm{cm}}^2$, is given by

$S=0.8r^2+\frac{1680}{r}$

making your method clear.

Figure 1 shows the plan view of a stage.

The plan view shows two congruent triangles $ABO$ and $GFO$ joined to a sector $OCDEO$ of a circle, centre $O$, where

• angle $COE=2.3$ radians

• arc length $CDE=27.6$ m

• $AOG$ is a straight line of length $15$ m

Show that $OC=12$ m.

Show that the size of angle $AOB$ is $0.421$ radians correct to $3$ decimal places.

Given that the total length of the front of the stage, $BCDEF$, is $35$ m, find the total area of the stage, giving your answer to the nearest square metre.

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

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$ where $x\in \mathrm{ℝ}$

Given that

• $\mathrm{f}\prime \left(x\right)=2x+\frac{1}{2}\cos x$

• the curve has a stationary point with $x$ coordinate $\alpha$

• $\alpha$ is small

use the small angle approximation for $\cos x$ to estimate the value of $\alpha$ to 3 decimal places.

Circle $C_1$ has equation $x^2+y^2=100$.

Circle $C_2$ has equation ${\left(x-15\right)}^2+y^2=40$.

The circles meet at points $A$ and $B$ as shown in Figure 3.

Show that angle $AOB$=0.635 radians to 3 significant figures, where $O$ is the origin.

The region shown shaded in Figure 3 is bounded by $C_1$ and $C_2$

Find the perimeter of the shaded region, giving your answer to one decimal place.

Figure 1 shows a plot of part of the curve with equation $y=\cos x$ where $x$ is measured in radians.

Diagram 1, shown below, is a copy of Figure 1.

Use Diagram 1 to show why the equation

$\cos x-2x-\frac{1}{2}=0$

has only one real root, giving a reason for your answer.

Given that the root of the equation is $\alpha$, and that $\alpha$ is small, use the small angle approximation for $\cos x$ to estimate the value of $\alpha$ to 3 decimal places.

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.