5.1 Radian measure
Degrees are an arbitrary choice — there is nothing special about \(360\). Radians tie the measurement of angle to the circle itself.
Definition 5.1. One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius.
The circumference of a circle of radius \(r\) is \(2\pi r\), which is \(2\pi \) radii, so a full revolution is \(2\pi \) radians. Hence \[2\pi \ \text {radians}=360^\circ ,\qquad \text {that is}\qquad \pi \ \text {radians}=180^\circ .\]
That single equation is the whole conversion. To go from degrees to radians multiply by \(\frac {\pi }{180}\); to go the other way multiply by \(\frac {180}{\pi }\).
Solution. Multiply each by \(\frac {\pi }{180}\) and cancel.
(a) \(45\times \frac {\pi }{180}=\frac {45\pi }{180}=\frac {\pi }{4}\).
(b) \(-30\times \frac {\pi }{180}=-\frac {30\pi }{180}=-\frac {\pi }{6}\).
(c) \(270\times \frac {\pi }{180}=\frac {270\pi }{180}=\frac {3\pi }{2}\).
Leave the answer as a multiple of \(\pi \) rather than converting to a decimal; that is the form every later formula expects.
Example 5.3. Express in degrees: (a) \(\frac {5\pi }{6}\), (b) \(\frac {3\pi }{2}\), (c) \(-\frac {7\pi }{4}\).
Solution. Multiply each by \(\frac {180}{\pi }\), so the \(\pi \) cancels.
(a) \(\frac {5\pi }{6}\times \frac {180}{\pi }=\frac {5(180)}{6}=150^\circ \).
(b) \(\frac {3\pi }{2}\times \frac {180}{\pi }=\frac {3(180)}{2}=270^\circ \).
(c) \(-\frac {7\pi }{4}\times \frac {180}{\pi }=-\frac {7(180)}{4}=-315^\circ \).
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