2.1: Radian Measure
While working on an experiment in your school science lab, your teacher asks you to turn up a detector by rotating the knob
Read this Concept, and at its conclusion, you will be able to turn the knob by the amount your teacher requested.
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Guidance
Until now, we have used degrees to measure angles. But, what exactly is a degree? A degree is
What if we were to rotate all the way around the circle? Continuing to add radius lengths, we find that it takes a little more than 6 of them to complete the rotation.
Recall from geometry that the arc length of a complete rotation is the circumference, where the formula is equal to
With this as our starting point, we can find the radian measure of other angles. Half of a rotation, or 180 degrees, must therefore be
Extending the radian measure past the first quadrant, the quadrantal angles have been determined, except
For the
Notice that the additional angles in the drawing all have reference angles of 45 degrees and their radian measures are all multiples of
Example A
Find the radian measure of these angles.
Angle in Degrees  Angle in Radians 

90 

45  
30 
Solution: Because 45 is half of 90, half of
and because 60 is twice as large as 30:
Here is the completed table:
Angle in Degrees  Angle in Radians 







There is a formula to convert between radians and degrees that you may already have discovered while doing this example. However, many angles that are commonly used can be found easily from the values in this table. For example, most students find it easy to remember 30 and 60. 30 is
Example B
Complete the following radian measures by counting in multiples of
Solution:
Notice that all of the angles with 60degree reference angles are multiples of
Example C
Find the radian measure of these angles.
Angle in Degrees  Angle in Radians 

120 

180  
240  
270  
300 
Solution: Because 30 is onethird of a right angle, multiplying gives:
adding this to the known value for ninety degrees of
Here is the completed table:
Angle in Degrees  Angle in Radians 



\begin{align*}180\end{align*}  \begin{align*}\pi\end{align*} 
\begin{align*}240\end{align*}  \begin{align*}\frac{4\pi}{3}\end{align*} 
\begin{align*}300\end{align*}  \begin{align*}\frac{5\pi}{3}\end{align*} 
Vocabulary
Radian: A radian (abbreviated rad) is the angle created by bending the radius length around the arc of a circle.
Guided Practice
1. Give the radian measure of \begin{align*}60^\circ\end{align*}
2. Give the radian measure of \begin{align*}75^\circ\end{align*}
3. Give the radian measure of \begin{align*}180^\circ\end{align*}
Solutions:
1. 30 is onethird of a right angle. This means that since \begin{align*}90^\circ = \frac{\pi}{2}\end{align*}, then \begin{align*}30^\circ = \frac{\pi}{6}\end{align*}. Therefore, multiplying gives:
\begin{align*}\frac{\pi}{6} \times 2 = \frac{\pi}{3}\end{align*}
2. 15 is onesixth of a right triangle. This means that since \begin{align*}90^\circ = \frac{\pi}{2}\end{align*}, then \begin{align*}15^\circ = \frac{\pi}{12}\end{align*}. Therefore, multiplying gives:
\begin{align*}\frac{\pi}{12} \times 5 = \frac{5\pi}{12}\end{align*}
3. Since \begin{align*}90^\circ = \frac{\pi}{2}\end{align*}, then \begin{align*}180^\circ = \frac{2\pi}{2} = \pi\end{align*}
Concept Problem Solution
Since \begin{align*}45^\circ = \frac{\pi}{4} rad\end{align*}, then \begin{align*}2 \times \frac{\pi}{4} = \frac{\pi}{2} = 2 \times 45^\circ\end{align*}. Therefore, a turn of \begin{align*}\frac{\pi}{2}\end{align*} is equal to \begin{align*}90^\circ\end{align*}, which is \begin{align*}\frac{1}{4}\end{align*} of a complete rotation of the knob.
Practice
Find the radian measure of each angle.
 \begin{align*}90^\circ\end{align*}
 \begin{align*}120^\circ\end{align*}
 \begin{align*}300^\circ\end{align*}
 \begin{align*}330^\circ\end{align*}
 \begin{align*}45^\circ\end{align*}
 \begin{align*}135^\circ\end{align*}
Find the degree measure of each angle.
 \begin{align*}\frac{3\pi}{2}\end{align*}
 \begin{align*}\frac{5\pi}{4}\end{align*}
 \begin{align*}\frac{7\pi}{6}\end{align*}
 \begin{align*}\frac{\pi}{6}\end{align*}
 \begin{align*}\frac{5\pi}{3}\end{align*}
 \begin{align*}\pi\end{align*}
 Explain why if you are given an angle in degrees and you multiply it by \begin{align*}\frac{\pi}{180}\end{align*} you will get the same angle in radians.
 Explain why if you are given an angle in radians and you multiply it by \begin{align*}\frac{180}{\pi}\end{align*} you will get the same angle in degrees.
 Explain in your own words why it makes sense that there are \begin{align*}2\pi\end{align*} radians in a circle.
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Image Attributions
Here you'll learn what radian measure is, and how to find the radian values for common angles on the unit circle.