While working on an experiment in your school science lab, your teacher asks you to turn up a detector by rotating the knob
Measure of Radians
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
For example,
Notice that the additional angles in the drawing all have reference angles of 45 degrees and their radian measures are all multiples of
Let's do some problems that involve radian measures.
1. Find the radian measure of these angles.
Angle in Degrees | Angle in Radians |
---|---|
90 | |
45 | |
30 |
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
2. Complete the following radian measures by counting in multiples of
Notice that all of the angles with 60-degree reference angles are multiples of
3. Find the radian measure of these angles.
Angle in Degrees | Angle in Radians |
---|---|
120 | |
180 | |
240 | |
270 | |
300 |
Because 30 is one-third 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*} |
Examples
Example 1
Earlier, you were given a problem about rotating the knob.
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.
Example 2
Give the radian measure of \begin{align*}60^\circ\end{align*}
30 is one-third 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*}
Example 3
Give the radian measure of \begin{align*}75^\circ\end{align*}
15 is one-sixth 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*}
Example 4
Give the radian measure of \begin{align*}180^\circ\end{align*}
Since \begin{align*}90^\circ = \frac{\pi}{2}\end{align*}, then \begin{align*}180^\circ = \frac{2\pi}{2} = \pi\end{align*}
Review
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.
Review (Answers)
To see the Review answers, open this PDF file and look for section 2.1.