Converting Between Radians and Degrees
Rationale
Often (particularly in science and engineering courses, and if you study calculus), the work that you do will deal with angles that are measured in radians (denoted rads). However, units of degrees come up in many applications, and most students are more familiar with these, so it is important to be able to convert between degrees and radians. In order to convert between degrees and radians we need to find an equation that relates the two.
Unit Conversion
Let's start with what we know. We know that the equation to determine the circumference of a circle is \(C =2\pi r\). We also know that the radius of the unit circle is 1. Therefore, the unit circle has a circumference of 2\(\pi\) . We also know that an entire circle is 360°. Therefore, we have:\[ 360^\circ = 2 \pi \text{ radians}\] or simply \[ 180^\circ= \pi \text{ radians}\]
Therefore, in order to convert between these two types of units, we have the
following two rules.
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To convert from degrees to radians, multiply by \(\frac{\pi}{180^\circ}\)
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To convert from radians to degrees. multiply by \(\frac{180^\circ}{\pi}\)
Example: Convert 30° to radians.
Solution: \[30^\circ \times \frac{\pi}{180^\circ} = \frac{30^\circ\pi}{180^\circ} = \frac{\pi}{6} \text{rads} \]
Example: Convert \(\frac{\pi}{3} \text{rads}\) to degrees.
Solution: \[ \frac{\pi}{3} \times \frac{180^\circ}{\pi} = \frac{180^\circ}{3} = 60^\circ\]
Example:
