When you teach unit circle trigonometry you've got to introduce radians, and inevitably that leads to problems. Students aren't initially comfortable with radians and you'll almost certainly have someone ask why they need to use radians.

To be sure, the degrees/minutes/seconds system only described a finite number of angles but with decimal degrees, there is the required infinite number of angles needed to map each degree to a point on the unit circle. For any angle in radians we can simply convert to degrees using the conversion factor $latex \frac{180^{\circ}}{\pi}$; why then do we need radians?

You can address this natural question by including a simple math model into your lesson. The displacement of a weight attached to a spring moves up and down over time (non damping) according to some equation like $latex y=\frac{1}{5}\cos(4t)$. Physclips is a good site for animations; check out this page to see spring movement juxtaposed with movement along the sine curve. You can download clips to your computer, the harmonic motion example is found here. After seeing how the equation models reality you're ready for the teaching point: **Radians are necessary because mathematical models use trig functions where the variable is time, which is not measured in degrees**.