Circular measure

What a radian is Paper 1

Degrees divide a full turn into 360 parts, a choice made for convenience thousands of years ago. A radian is a natural unit instead, built from the circle itself:

one radian is the angle at the centre of a circle subtended by an arc whose length equals the radius.

The whole circumference is $2\pi r$, which is $2\pi$ radii, so a full turn is $2\pi$ radians:

$$2\pi \text{radians}={360}^{°},\pi \text{radians}={180}^{°}$$

So 1 radian is $\frac{{180}^{°}}{\pi }\approx {57.3}^{°}$.

Converting:

• degrees to radians: multiply by $\frac{\pi }{180}$; for example ${135}^{°}=135\times \frac{\pi }{180}=\frac{3\pi }{4}$;
• radians to degrees: multiply by $\frac{180}{\pi }$; for example $\frac{5\pi }{6}=\frac{5\times {180}^{°}}{6}={150}^{°}$.

Learn the common ones, which appear as exact values throughout Pure Mathematics: ${30}^{°}=\frac{\pi }{6}$, ${45}^{°}=\frac{\pi }{4}$, ${60}^{°}=\frac{\pi }{3}$, ${90}^{°}=\frac{\pi }{2}$, ${180}^{°}=\pi$, ${270}^{°}=\frac{3\pi }{2}$.

When no unit is shown, an angle is in radians: "$\theta =1.2$" means 1.2 radians. Set your calculator to radian mode for this topic, and check it before every trigonometric calculation: $\sin 1.2$ is $0.932$ in radians but $0.0209$ in degrees.

Arc length and sector area Paper 1

A sector is the region between two radii and the arc joining them. If the angle at the centre is $\theta$ radians, the sector is the fraction $\frac{\theta }{2\pi }$ of the whole circle. So:

• arc length: $s=\frac{\theta }{2\pi }\times 2\pi r=r\theta$;
• sector area: $A=\frac{\theta }{2\pi }\times \pi r^2=\frac{1}{2}{r}^2\theta$.

These simple formulae are the reason radians are used: in degrees the same results carry an untidy $\frac{\pi }{180}$, as $s=\frac{\theta }{360}\times 2\pi r$. Both formulae need $\theta$ in radians. If an angle is given in degrees, convert it first.

The perimeter of a sector is two radii plus the arc: $2r+r\theta$. A very common error is to give the arc length alone.

The formulae can be used backwards. If you know any two of $r$, $\theta$, $s$ and $A$, you can find the others: for example $A=\frac{1}{2}{r}^2\theta =\frac{1}{2}r(r\theta )=\frac{1}{2}rs$, so an area and an arc length together give $r$.

Questions often ask for exact answers: if $\theta$ is a multiple of $\pi$, leave $\pi$ in the answer, for example an arc length of $4\pi$ cm.

The perimeter of the sector is the two straight edges $OA$ and $OB$ plus the curved arc $AB$.

Segments, triangles and composite shapes Paper 1

Many questions combine sectors with triangles, and the syllabus expects the trigonometry of triangles from IGCSE: right-angled trigonometry, the sine rule, the cosine rule and $\text{area}=\frac{1}{2}ab\sin C$.

A segment is the region between a chord and its arc. Its area is the sector minus the triangle:

$$\text{segment area}=\frac{1}{2}{r}^2\theta -\frac{1}{2}{r}^2\sin \theta =\frac{1}{2}{r}^2(\theta -\sin \theta )$$

because the triangle $OAB$ has two sides $r$ with the angle $\theta$ between them. With $\theta$ in radians, use radian mode for $\sin \theta$.

The chord $AB$ can be found in two ways:

• the cosine rule: $A{B}^2=r^2+r^2-2r^2\cos \theta$;
• splitting the isosceles triangle $OAB$ down its line of symmetry into two right-angled triangles: $AB=2r\sin \frac{\theta }{2}$.

The perimeter of a segment is the chord plus the arc: $AB+r\theta$.

Tangents. A tangent meets the radius at ${90}^{°}$, so a sector with a tangent drawn at one end gives a right-angled triangle, and $\tan$ links the tangent length to the radius: $AT=r\tan \theta$ when angle $AOT=\theta$.

Method for composite shapes: mark every length and angle you know on the diagram, find the missing lengths of the triangles with trigonometry, and build the required area or perimeter from sectors and triangles, added or subtracted. Keep exact values ($\sqrt{3}$, $\pi$) until the end if the answer must be exact, and full calculator values otherwise.

The shaded segment is the sector $OAB$ with the triangle $OAB$ taken away.