Converting Degrees and Radians

There are two different systems for measuring angles: degrees and radians. To measure in degrees, we first consider a full revolution as $360^\circ$. Then by evenly dividing the revolution into 360 parts, each of these parts is $1^\circ$ and for any integer $1 \leq d \leq 360,$ the measure of $d$ combined parts gives $d^\circ.$ The value 360 satisfies the nice property that it has many divisors $(2, 3, 4, 5, 6, 8, 10, 12, 15, \ldots).$

The second system of measurement is by radians, in which we measure an angle by measuring the arc length of the unit circle. Consider a unit circle centered at the origin and let $\theta$ denote an angle formed with the positive $x$-axis.

Then the measure of angle $\theta$ in radians is the arc length of the circle carved out by the angle. To see how to compute this, let's start with a few easy examples. If the angle measures $0^\circ$ in degrees, then the arc length of the circle carved out by the angle is $0$. On the other hand, if the angle measures $360^\circ$ in degrees, then the arc length of the circle carved out by the angle is the circumference of a circle with radius $1$, or $2 \pi \cdot 1 = 2\pi.$ For any other angle measurement in degrees, we can calculate the fraction of the arc length carved out by converting from degrees to radians, as shown in the following example:

What is the measure of a right angle in radians?

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Since a right angle is $90^\circ$ in degrees, we have

$$90^\circ = 90^\circ \times \frac{\pi \text{ radians} }{180^\circ} = \frac{\pi }{2} \text{ radians}. \ _\square$$

Converting between degrees and radians:

• To convert a measure in degrees to radians, multiply by $\frac{\pi \text{ radians}}{180^\circ}$.
• To convert a measure in radians to degrees, multiply by $\frac{180^\circ}{\pi \text{ radians}}$.

Another simple way to convert the degrees into radians is to use the following formula: $$\frac{D}{90}=\frac{\ \ R\ \,}{\frac{\pi}{2}}.$$ This is true because the numbers of radians in $90^\circ$ is equal to $\dfrac{\pi}{2}$ radians. So, we can equate the expressions and derive this expression.

Here are a few radian measures that are useful to know:

$$\begin{array}{c}&30^\circ = \frac{\pi }{6} \text{ radians}, &60^\circ = \frac{\pi }{3} \text{ radians}, &90^\circ = \frac{\pi }{2} \text{ radians}, &180^\circ = \pi \text{ radians}. \end{array}$$

For a general circle with radius $r$, if an angle $\theta$ (given in radians) is formed with the positive $x$-axis, then the arc length of the portion of the circle carved out by the angle is

$$\text{(Arc Length)} = r \theta.$$

As we have seen, for a circle with radius $r=1,$ the arc length is simply the angle $\theta$ in radians.

Convert the following from radians to degrees:$$$$

$$\pi \text{ radians}, ~ \frac{\pi}{2} \text{ radians}.$$

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Since $1 \text{ radian} = \frac{360^\circ}{2\pi} ,$ we have

$$\begin{aligned} \pi \text{ radians} &= \pi \text{ radians} \cdot \frac{360^{\circ}}{2\pi \text{ radians}} \\ &= 180^\circ. \end{aligned}$$

Similarly,

$$\begin{aligned} \frac{\pi}{2} \text{ radians} &= \frac{\pi}{2} \text{ radians} \cdot \frac{360^{\circ}}{2\pi \text{ radians}} \\ &= 90^\circ. \ _\square \end{aligned}$$

Convert the following from degrees to radians:$$$$

$$60^\circ, ~180^\circ.$$

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Since $2 \pi \text{ radians} = \frac{360^\circ}{2\pi} ,$ we have

$$\begin{aligned} 60^\circ &= 60^\circ \cdot \frac{2 \pi \text{ radians}}{360^\circ} \\ &= \frac{\pi}{3} \text{ radians} \\ \\ 180^\circ &= 180 \cdot \frac{2 \pi \text{ radians}}{360^\circ} \\ &= \pi \text{ radians}. \ _\square \end{aligned}$$

What is the central angle $\theta$ (in radians) that subtends an arc of length 4 and radius 2?

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Approach 1:

When an arc length equals its radius, then the central angle that subtends the arc is 1 radian.
Since the arc length of the circle is twice the radius in this problem, the central angle is 2 radians. $_\square$

Approach 2:

The length $l$ of an arc of radius $r$ and central angle $\theta$ is given by the formula $$l=r\theta.$$ For this problem we have \[ \begin{align} l &= r\theta \\ 4 &= 2\theta \\ \theta &= 2 \text{ radians}.\ _ \square
\end{align} \]