<< Chapter < Page | Chapter >> Page > |
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, $\text{\hspace{0.17em}}\mathrm{360\xb0}.$ 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.
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,
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.
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 $\text{\hspace{0.17em}}\pi .$
To convert between degrees and radians, use the proportion
Convert each radian measure to degrees.
Because we are given radians and we want degrees, we should set up a proportion and solve it.
Convert $\text{\hspace{0.17em}}-\frac{3\pi}{4}\text{\hspace{0.17em}}$ radians to degrees.
$\mathrm{-135}\xb0$
Notification Switch
Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?