Once you’ve mastered the unit circle, how do you extend your knowledge to trig functions over many cycles?

The unit circle looks  like this:

 

The numbers around the circle represent angles, measured in radians. The angle variable is called theta.

How can you represent that angle on a coordinate plane? Imagine that you break the unit circle open at theta = 0,  open it, and straighten it into a line.

That line is an x-axis. Add a y-axis and you have a coordinate plane set up to graph trig functions.

Thanks to Asma Stanikzai, a student in MTH167, for this idea.