Topic 15 Circular measure
15.1 Radians and degrees
Angles are normally measured in degrees, e.g. a right angle has \(90^\circ\). In mathematics and engineering it is more natural and convenient to measure angles in radians: the length of an arc of a unit circle is equal to the measurement, in radians, of the angle that it subtends.
Figure 15.1: Unit circle and radians
| degrees | \(360\) | \(180\) | \(90\) | \(60\) | \(45\) | \(30\) | \(0\) |
| radians | \(2\pi\) | \(\pi\) | \(\frac{\pi}{2}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{6}\) | \(0\) |
15.1.1 Converting degrees to radians
\[\theta^\circ = \left(\dfrac{2\pi}{360}\cdot\theta\right)\,\mathrm{rad}\]
Example 15.1 Find the value of \(23^\circ\) in radians.
Solution. \(23^\circ = \left(\dfrac{2\pi}{360}\cdot 23\right)\,\mathrm{rad} \approx 0.4\,\mathrm{rad}\).
15.1.2 Converting radians to degrees
\[\theta\,\mathrm{rad}=\left(\dfrac{360}{2\pi}\cdot\theta\right)^\circ\]
Example 15.2 Find the value of \(1.5\,\mathrm{rad}\) in degrees.
Solution. \(1.5\,\mathrm{rad} = \left(\dfrac{360}{2\pi}\cdot 1.5\right)^\circ = \left(\dfrac{270}{\pi}\right)^\circ \approx 85.9^\circ\).
15.2 Arc and sector
15.2.1 Length of an arc
The length of an arc of a circle is given by \(s=r\cdot\theta\), where \(r\) is the radius and \(\theta\) is the angle subtended in radians.
Example 15.3 Determine the length \(s\) of an arc of a circle with radius \(45mm\) when the angle subtended is \(85^\circ\).
Solution. \(s=r\theta = 45\,\mathrm{mm}\cdot\left(\dfrac{2\pi}{360}\cdot 85\right) \approx 66.8\,\mathrm{mm}\).
15.2.2 Area of a sector
The area of a sector of a disc is \(A=\left(\dfrac{\theta}{2\pi}\cdot\pi\cdot r^2\right) = \dfrac{1}{2}\cdot r^2\cdot \theta\), where \(r\) is the radius of \(\theta\) is the angle subtended in radians.
Example 15.4 A light source spreads illumination through an angle \(130^\circ\) to a distance of \(35m\). Determine the illuminated area \(A\).
Solution. \(A=\frac12 r^2\theta = \frac12\left(35\,\mathrm{m}\right)^2\cdot\left(\dfrac{2\pi}{360}\cdot 130\right) \approx 1390\,\mathrm{m}^2\).