radian

A dimensionless unit of measure of angles. Although the ancient Babylonian ‘degree’ unit of angle measure is still in wide use, in mathematics we prefer to use the radian measure. Given a circle centered at the origin in the Cartesian plane, imagine taking a flexible line segment equal in length to the radius and laying it along the outside of the circle, beginning at the horizontal axis and going counterclockwise.

Figure 1: An angle of 1 radian.

This arc subtends an angle of one radian. Because the circumference of a circle is twice the radius times \(\pi\), a full circle corresponds to an angle of \(2\pi\) radians. By identifying 360° with \(2\pi\) radians we can easily derive these many correspondences between degree measure and radian measure:

Figure 2: Radian and degree correspondences.

It is never difficult to convert any arbitrary degree measure to radians or vice versa if you remember this simple relationship:

\[\frac{\mbox{radians}}{\mbox{degrees}}=\frac{\pi}{180}\]

By remembering the correspondence between radians and degrees indicated by the formula above one may always convert radians to degrees and vice versa by plugging the known quantity into the equation and solving for the unknown quantity. To convert from degrees to radians, multiply by \(\pi\) and divide by 180. To convert from radians to degrees, divide by \(\pi\) and multiply by 180. With practice, using radian measure becomes as natural as using degrees, and the use of radians greatly simplifies our work with the trigonometric functions, especially in the calculus.