## Why radians? The use of 360$\textdegree$ to denote a full revolution of a circle seems somewhat arbitrary and is likely to be a [historical accident](https://www.scienceabc.com/pure-sciences/why-is-a-full-circle-360-degrees-instead-of-something-more-convenient-like-100.html) caused by mismeasurement of the number of days in the year that was then retained because 360 is highly divisible and therefore convenient. That said, a purer form is possible which does not rely on the choice of an arbitrary number of divisions and that is *radians*. Angles in radians are commonly denoted either by the suffix "$\text{rad}quot; or by leaving them unitless. You can generally deduce from context if radians are being used if you see no units but the angles have a sprinkling of $\pi$ about them. Apparently some old texts use $^r$ (superscript $r$) although that seems likely to cause confusion given $r$ is a commonly-used variable name. ## Definition of Radians Consider the unit circle in the diagram below ![[Unit Circle Radians.png]] What is the circumference of this circle? $2 \pi r$ is the circumference of circles in general, and since the radius is 1 (it is a unit circle), the circumference is simply $2\pi$ in this case. When measuring angles in radians, we simply name the angle after the length of the arc on the circumference of a unit circle that subtends that angle. That is, in the diagram above, if the length of the arc that subtends the angle $\angle CAB$ is $r$ and $m \angle CAB = \theta$, then $\theta = r \text{ rad}$. This allows a pure definition of angles in terms of the unit circle without any additional scaling or divisor constant. It follows that $2\pi \text{ rad} = 360\textdegree$ and $\pi \text{ rad} = 180\textdegree$, which is the main factor used for conversions. ## Common angles in degrees and radians | angle in $\textdegree$ | angle in radians | | :----------------------: | : ----------------: | | 360$\textdegree$ | $2 \pi \text{ rad}$ | | 180$\textdegree$ | $\pi \text{ rad}$ | | 90$\textdegree$ | $\frac{\pi}{2} \text{ rad}$ | | 60$\textdegree$ | $\frac{\pi}{3} \text{ rad}$ | | 45$\textdegree$ | $\frac{\pi}{4} \text{ rad}$ | | 30$\textdegree$ | $\frac{\pi}{6} \text{ rad}$ | ## Converting arbitrary quantities between degrees and radians As stated above, in order to do conversions, the factor required is $ \pi \text{ rad} = 180\textdegree$ It follows that the ratios to bear in mind are $\frac{\pi \text{ rad}}{180\textdegree}$ and $\frac{180\textdegree}{\pi \text{ rad}}$. If you're concerned about getting these the wrong way round, you can simply avoid errors by always writing the units as part of your calculation; if you are using the correct ratio the units will cancel. ### Worked examples 1. Convert $\frac{\pi}{9} \text{ rad}$ to degrees $ \begin{align*} x &= \frac{\pi}{9} \text{ rad} \cdot \frac{180\textdegree}{\pi \text{ rad}} \\ &= \frac{180}{9} \\ &= 20\textdegree \end{align*} $ 2. Convert 120$\textdegree$ to radians. $ \begin{align*} x &= 120\textdegree \cdot \frac{\pi \text{ rad}}{180\textdegree} \\ &= \frac{120 \pi}{180} \; \text{rad} \\ &= \frac{2 \pi}{3} \; \text{rad} \end{align*} $ As can be seen, it is commonly expected to give radians as simplified fractions in terms of $\pi$. If there is a surd in the denominator, it is conventional to "rationalize" the fraction (multiply top and bottom by this in order to move it to the numerator). #### Exercises Chapters 1 & 2 of Chris McMullen's trig workbook give more conversion exercises than you could possibly hope for to get fluent in these.