4.1 Radians and Degrees

So far we have been using degrees as our unit of measurement for angles. However, there is another way of measuring angles that is often more convenient. The idea is simple: associate a central angle of a circle with the arc that it intercepts.

Consider a circle of radius $r>0$, as in Figure 4.1.1. In geometry you learned that the circumference $C$ of the circle is $C = 2\;\pi\;r$, where $\pi = 3.14159265...$.

tikzpicture[every node/.style=font=] (0,0) circle (1.2); (0,0) circle (2pt); [below] at (0,0) O; (0,0) -- (0:1.2); [right] at (0:1.2) A; (0,0) -- (90:1.2); [above] at (90:1.2) B; [linecolor,line width=1.5pt] (0:1.2) arc (0:90:1.2); (0:1.2) circle (2pt); (90:1.2) circle (2pt); [above] at (90:1.6) AB = 14C = 2\,r; at (45:0.4) 90 ; [dashed,-latex] (0:0.7) arc (0:90:0.7); tikzpicture — Figure 4.1.1Angle $\theta$ and intercepted arc $\wideparen{AB}$ on circle of circumference $C=2\pi r$

tikzpicture[every node/.style=font=] (0,0) circle (1.2); (0,0) circle (2pt); [below] at (0,0) O; (0,0) -- (0:1.2); [right] at (0:1.2) A; (0,0) -- (180:1.2); [left] at (180:1.2) B; [linecolor,line width=1.5pt] (0:1.2) arc (0:180:1.2); (0:1.2) circle (2pt); (180:1.2) circle (2pt); [above] at (90:1.6) AB = 12C = \;r; at (90:0.4) 180 ; [dashed,-latex] (0:0.7) arc (0:180:0.7); tikzpicture — Figure 4.1.1Angle $\theta$ and intercepted arc $\wideparen{AB}$ on circle of circumference $C=2\pi r$

In Figure 4.1.1 we see that a central angle of $90\Degrees$ cuts off an arc of length $\tfrac{\pi}{2}\,r$, a central angle of $180\Degrees$ cuts off an arc of length $\pi\,r$, and a central angle of $360\Degrees$ cuts off an arc of length $2\pi\,r$, which is the same as the circumference of the circle. So associating the central angle with its intercepted arc, we could say, for example, that

\begin{displaymath} 360\Degrees \quad\text{``equals''}\quad 2\pi\,r \quad\text{(or $2\pi$ `radiuses').} \end{displaymath}

The radius $r$ was arbitrary, but the $2\pi$ in front of it stays the same. So instead of using the awkward “radiuses” or “radii”, we use the term radians:

\begin{equation} \boxed{360\Degrees ~=~ 2\pi ~~\text{radians}} \tag{4.1}\end{equation}

The above relation gives us any easy way to convert between degrees and radians:

\begin{alignat}{3} \text{\textbf{Degrees to radians:}}&\quad x~~\text{degrees}\quad&=\quad \left( \frac{\pi}{180} \;\cdot\; x \right) ~~\text{radians}\tag{4.2}\\ \text{\textbf{Radians to degrees:}}&\quad x~~\text{radians}\quad&=\quad \left( \frac{180}{\pi} \;\cdot\; x \right) ~~\text{degrees} \tag{4.3}\end{alignat}

Formula (4.2) follows by dividing both sides of equation (4.1) by $360$, so that $1\Degrees = \frac{2\pi}{360} = \frac{\pi}{180}$ radians, then multiplying both sides by $x$. Formula (4.3) is similarly derived by dividing both sides of equation (4.1) by $2\pi$ then multiplying both sides by $x$.

The statement $\theta = 2\pi$ radians is usually abbreviated as $\theta = 2\pi$ rad, or just $\theta = 2\pi$ when it is clear that we are using radians. When an angle is given as some multiple of $\pi$, you can assume that the units being used are radians.

Example 4.1

Convert $18\Degrees$ to radians.

Solution: Using the conversion formula (4.2) for degrees to radians, we get

\begin{displaymath} 18\Degrees ~=~ \frac{\pi}{180} \;\cdot\; 18 ~=~ \boxed{\frac{\pi}{10} ~~\text{rad}} ~. \end{displaymath}

Example 4.2

Convert $\frac{\pi}{9}$ radians to degrees.

Solution: Using the conversion formula (4.3) for radians to degrees, we get

\begin{displaymath} \frac{\pi}{9} ~~\text{rad} ~=~ \frac{180}{\pi} \;\cdot\; \frac{\pi}{9} ~=~ \boxed{20\Degrees} ~. \end{displaymath}

Table 4.1Commonly used angles in radians

2.0

Degrees	Radians	Degrees	Radians	Degrees	Radians	Degrees	Radians
$0\Degrees$	$0$	$90\Degrees$	$\dfrac{\pi}{2}$	$180\Degrees$	$\pi$	$270\Degrees$	$\dfrac{3\pi}{2}$
$30\Degrees$	$\dfrac{\pi}{6}$	$120\Degrees$	$\dfrac{2\pi}{3}$	$210\Degrees$	$\dfrac{7\pi}{6}$	$300\Degrees$	$\dfrac{5\pi}{3}$
$45\Degrees$	$\dfrac{\pi}{4}$	$135\Degrees$	$\dfrac{3\pi}{4}$	$225\Degrees$	$\dfrac{5\pi}{4}$	$315\Degrees$	$\dfrac{7\pi}{4}$
$60\Degrees$	$\dfrac{\pi}{3}$	$150\Degrees$	$\dfrac{5\pi}{6}$	$240\Degrees$	$\dfrac{4\pi}{3}$	$330\Degrees$	$\dfrac{11\pi}{6}$

tikzpicture[every node/.style=font=] (0,0) circle (1.2); (0,0) circle (2pt); [below] at (0,0) O; (0,0) -- (0:1.2) node[below,midway] r; (0,0) -- (57.3:1.2); [linecolor,line width=1.5pt] (0:1.2) arc (0:57.3:1.2); (0:1.2) circle (2pt); (57.3:1.2) circle (2pt); [above right] at (28:1.2) r; at (28:0.4) ; [dashed,-latex] (0:0.7) arc (0:57.3:0.7); [below] at (270:1.2) = 1 radian; tikzpicture — Figure 4.1.2

Table 4.1 shows the conversion between degrees and radians for some common angles. Using the conversion formula (4.3) for radians to degrees, we see that

\begin{displaymath} 1 ~~\text{radian} ~~=~~ \frac{180}{\pi}~~\text{degrees} ~~\approx~~ 57.3\Degrees ~. \end{displaymath}

Formally, a radian is defined as the central angle in a circle of radius $r$ which intercepts an arc of length $r$, as in Figure 4.1.2. This definition does not depend on the choice of $r$ (imagine resizing Figure 4.1.2). One reason why radians are used is that the scale is smaller than for degrees. One revolution in radians is $2\pi \approx 6.283185307$, which is much smaller than $360$, the number of degrees in one revolution. The smaller scale makes the graphs of trigonometric functions (which we will discuss in Chapter 5) have similar scales for the horizontal and vertical axes. Another reason is that often in physical applications the variables being used are in terms of arc length, which makes radians a natural choice.

The default mode in most scientific calculators is to use degrees for entering angles. On many calculators there is a button labeled 2pt DRG for switching between degree mode (D), radian mode (R), and gradian mode (G).^[1] On some graphing calculators, such as the the TI-83, there is a 2pt MODE button for changing between degrees and radians. Make sure that your calculator is in the correct angle mode before entering angles, or your answers will likely be way off. For example,

\begin{align*} \sin\;4\Degrees ~&=~ \phantom{-}0.0698 ~,\\ \sin\;(4~\text{rad}) ~&=~ -0.7568 ~, \end{align*}

so the values are not only off in magnitude, but do not even have the same sign. Using your calculator’s 1pt $\sin^{-1}$, 1pt $\cos^{-1}$, and 1pt $\tan^{-1}$ buttons in radian mode will of course give you the angle as a decimal, not an expression in terms of $\pi$.

You should also be aware that the math functions in many computer programming languages use radians, so you would have to write your own angle conversions.^[2]

Exercises

For Exercises 1-5, convert the given angle to radians.

$4\Degrees$
$15\Degrees$
$130\Degrees$
$275\Degrees$
$-108\Degrees$

For Exercises 6-10, convert the given angle to degrees.

$4$ rad
$\dfrac{\pi}{5}$ rad
$\dfrac{11\pi}{9}$ rad
$\dfrac{29\pi}{30}$ rad
$35$ rad
Put your calculator in radian mode and take the cosine of $0$. Whatever the answer is, take its cosine. Then take the cosine of the new answer. Keep repeating this. On most calculators after about $50$-$60$ iterations you should start to see the same answer repeating. What is that number? Try starting with a number different from $0$. Do you get the same answer repeating after roughly the same number of iterations as before? Try the same procedure in degree mode, starting with $0\Degrees$. Does the same thing happen? If so, does it take fewer iterations for the answer to start repeating than in radian mode, or more?

A gradian is defined as $\tfrac{1}{400}$ of a circle, i.e. there are $400$ gradians in one revolution. Compared to the more common $360\Degrees$ in one revolution, gradians appear to be easier to work with, since a right angle is $100$ gradians (thus making integer multiples of a right angle easier to remember). Outside of a few specialized areas (e.g. artillery calculations), gradians are nevertheless not widely used today. ↩
One exception is Octave, which has functions cosd(), sind(), tand() that take angles in degrees as parameters, in addition to the usual cos(), sin(), tan() functions which use radians. ↩

Degrees	Radians	Degrees	Radians	Degrees	Radians	Degrees	Radians
\(0\Degrees\)	\(0\)	\(90\Degrees\)	\(\dfrac{\pi}{2}\)	\(180\Degrees\)	\(\pi\)	\(270\Degrees\)	\(\dfrac{3\pi}{2}\)
\(30\Degrees\)	\(\dfrac{\pi}{6}\)	\(120\Degrees\)	\(\dfrac{2\pi}{3}\)	\(210\Degrees\)	\(\dfrac{7\pi}{6}\)	\(300\Degrees\)	\(\dfrac{5\pi}{3}\)
\(45\Degrees\)	\(\dfrac{\pi}{4}\)	\(135\Degrees\)	\(\dfrac{3\pi}{4}\)	\(225\Degrees\)	\(\dfrac{5\pi}{4}\)	\(315\Degrees\)	\(\dfrac{7\pi}{4}\)
\(60\Degrees\)	\(\dfrac{\pi}{3}\)	\(150\Degrees\)	\(\dfrac{5\pi}{6}\)	\(240\Degrees\)	\(\dfrac{4\pi}{3}\)	\(330\Degrees\)	\(\dfrac{11\pi}{6}\)