7.1 Angles  (Page 4/29)

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Because radian    measure is the ratio of two lengths, it is a unitless measure. For example, in [link] , suppose the radius were 2 inches and the distance along the arc were also 2 inches. When we calculate the radian measure of the angle, the “inches” cancel, and we have a result without units. Therefore, it is not necessary to write the label “radians” after a radian measure, and if we see an angle that is not labeled with “degrees” or the degree symbol, we can assume that it is a radian measure.

Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, $\text{\hspace{0.17em}}360°.$ We can also track one rotation around a circle by finding the circumference, $\text{\hspace{0.17em}}C=2\pi r,$ and for the unit circle $\text{\hspace{0.17em}}C=2\pi .\text{\hspace{0.17em}}$ These two different ways to rotate around a circle give us a way to convert from degrees to radians.

Identifying special angles measured in radians

In addition to knowing the measurements in degrees and radians of a quarter revolution, a half revolution, and a full revolution, there are other frequently encountered angles in one revolution of a circle with which we should be familiar. It is common to encounter multiples of 30, 45, 60, and 90 degrees. These values are shown in [link] . Memorizing these angles will be very useful as we study the properties associated with angles.

Now, we can list the corresponding radian values for the common measures of a circle corresponding to those listed in [link] , which are shown in [link] . Be sure you can verify each of these measures.

Find the radian measure of one-third of a full rotation.

For any circle, the arc length along such a rotation would be one-third of the circumference. We know that

So,

$\begin{array}{ccc}\hfill s& =& \frac{1}{3}\left(2\pi r\right)\hfill \\ & =\hfill & \frac{2\pi r}{3}\hfill \end{array}$

The radian measure would be the arc length divided by the radius.

Find the radian measure of three-fourths of a full rotation.

$\frac{3\pi }{2}$

Because degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion where $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is the measure of the angle in degrees and $\text{\hspace{0.17em}}{\theta }_{R}\text{\hspace{0.17em}}$ is the measure of the angle in radians.

$\frac{\theta }{180}=\frac{{\theta }_{{}^{R}}}{\pi }$

This proportion shows that the measure of angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in degrees divided by 180 equals the measure of angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians divided by $\text{\hspace{0.17em}}\pi .\text{\hspace{0.17em}}$ Or, phrased another way, degrees is to 180 as radians is to $\text{\hspace{0.17em}}\pi .$

$\frac{\text{Degrees}}{180}=\frac{\text{Radians}}{\pi }$

To convert between degrees and radians, use the proportion

$\frac{\theta }{180}=\frac{{\theta }_{R}}{\pi }$

Convert each radian measure to degrees.

1. $\frac{\pi }{6}$
2. 3

Because we are given radians and we want degrees, we should set up a proportion and solve it.

1. We use the proportion, substituting the given information.
$\begin{array}{ccc}\hfill \frac{\theta }{180}& =& \frac{{\theta }_{R}}{\pi }\hfill \\ \hfill \frac{\theta }{180}& =& \frac{\frac{\pi }{6}}{\pi }\hfill \\ \hfill \theta & =& \frac{180}{6}\hfill \\ \hfill \theta & =& 30°\hfill \end{array}$
2. We use the proportion, substituting the given information.
$\begin{array}{ccc}\hfill \frac{\theta }{180}& =& \frac{{\theta }_{{}^{R}}}{\pi }\hfill \\ \hfill \frac{\theta }{180}& =& \frac{3}{\pi }\hfill \\ \hfill \theta & =& \frac{3\left(180\right)}{\pi }\hfill \\ \hfill \theta & \approx & 172°\hfill \end{array}$

Convert $\text{\hspace{0.17em}}-\frac{3\pi }{4}\text{\hspace{0.17em}}$ radians to degrees.

$-135°$

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