Using Radians

Because radian measure is the ratio of two lengths, it is a unitless measure. For example, 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.
\( \require{cancel}\)
\[\theta \, \text{radians} = \frac{s}{r} =  \frac{2 \cancel{\text{in.}}}{2 \cancel{\text{in.}}} = 1 \]
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, 360°. We can also track one rotation around a circle by finding the circumference, C=2πr, and for the unit circle C=2π. These two different ways to rotate around a circle give us a way to convert from degrees to radians.
\[ \begin{align}
1\, \text{rotation} &= 360^\circ = 2\pi \,\text{radians} \\
 \frac{1}{2}\, \text{rotation} &= 180^\circ = \pi \,\text{radians} \\
 \frac{1}{4}\, \text{rotation} &= 90^\circ = \frac{\pi}{2} \,\text{radians}
\end{align}\]
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. Memorizing these angles will be very useful as we study the properties associated with angles. Here, we can list the corresponding radian values for the common measures.

YouTube Video: Radian Measure

Attribution: OpenStax CAT, Algebra and Trigonometry. OpenStax CNX. Aug 11, 2015 http://cnx.org/contents/13ac107a-f15f-49d2-97e8-60ab2e3b519c@5.189.