\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Appendix A: Miscellany
11.

Degrees and Radians

In everyday life, angles are usually measured in degrees: a right angle is 90°, an equilateral triangle has three 60° angles, a whole circle is 360°. The number 360 is nice because it is divisible by so many other numbers.

Mathematicians and physicists, however, often like to measure angles in radians, rather than degrees. A radian is defined so that $$360^\circ = 2\pi \u{rad} \quad \hbox{or} \quad 1\u{rad} \approx 57.3^\circ$$ Why would we ever want to use a unit as clumsy as that?

(no alternate text)

For one very good reason. A circle with radius $R$ has a circumference of $s=2\pi R$. Now consider a semicircle with the same radius: the length of the curved part of the semicircle is half of this, or $s=\pi R$. (The "length of the curved part" is called the arc length of the semicircle.) If we cut the circle into sixths, the arc-length of one of those slices is $s={2\pi\over 6}R$ or $\frac13\pi R$.

But what is the angle $\theta$ of that slice? Well, if a complete circle is $2\pi\u{rad}$ around, then one-sixth of the way around should be $2\pi\over 6={\pi\over 3}$, which is exactly the coefficient in the arc-length expression. This would work no matter how we slice the circle, and so we have the expression

$$s=r\theta, \text{but only if $\theta$ is in radians}$$

There are some cases where knowing the arc-length could be very useful indeed; for instance, if you're thinking about the distance a car travels around a curve. Therefore, while we will usually use degrees when talking about angles, occasionally we will need to break out the radians. How can you tell the difference? Degrees will always have a ° symbol after their name, for one, while radians will often be tagged with $\u{rad}.^\ast$ Generally speaking, if you see a $\pi$ in an angle, then it's probably in radians, although that's not a guarantee, and the reverse is less often true.

Footnote

${}^\ast$Technically speaking, angles are dimensionless: if we solve for $\theta$ in the above equation, we get $\theta={s\over r}$, and since $s$ and $r$ are both lengths, the meters cancel, and we're left with no units. The unit "radians" is mostly used as a flag to tell us that this is an angle in radians, but sometimes it will just disappear during a calculation. For instance, when using $s=r\theta$ for a circle with radius $r=0.2\u{m}$ and an angle of $\theta=\pi\u{rad}$: $$s=r\theta = (0.2\u{m})(\pi\u{rad}) = 0.2\u{m}, \, (\textit{not }\u{m\cdot rad}).$$