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Subsection B.2.1 Angles — Radians vs Degrees

For mathematics, and especially in calculus, it is much better to measure angles in units called radians rather than degrees. By definition, an arc of length \(\theta\) on a circle of radius one subtends an angle of \(\theta\) radians at the centre of the circle.

The circle on the left has radius 1, and the arc swept out by an angle of \(\theta\) radians has length \(\theta\text{.}\) Because a circle of radius one has circumference \(2\pi\) we have

\begin{align*} 2\pi\text{ radians }&=360^\circ & \pi\text{ radians }&=180^\circ & \frac{\pi}{2}\text{ radians }&=90^\circ\\ \frac{\pi}{3}\text{ radians }&=60^\circ & \frac{\pi}{4}\text{ radians }&=45^\circ & \frac{\pi}{6}\text{ radians }&=30^\circ \end{align*}

More generally, consider a circle of radius \(r\text{.}\) Let \(L(\theta)\) denote the length of the arc swept out by an angle of \(\theta\) radians and let \(A(\theta)\) denote the area of the sector (or wedge) swept out by the same angle. Since the angle sweeps out the fraction \(\frac{\theta}{2\pi}\) of a whole circle, we have

\begin{align*} L(\theta) &= 2\pi r \cdot \frac{\theta}{2\pi} = \theta r & \text{and}\\ A(\theta) &= \pi r^2 \cdot \frac{\theta}{2\pi} = \frac{\theta}{2} r^2 \end{align*}
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