The Tangent is a Tangent!

In my post Trigonometry Yoga, I discussed how defining sine and cosine as lengths of segments in a unit circle helps develop intuition for these functions.

I learned the circle definitions of sine and cosine in my junior year of high school, in the class that would now be called pre-calculus (it was called “Trig Senior Math”). Two years earlier, I’d learned the triangle definitions of sine, cosine, and tangent in geometry class. I don’t remember any of my teachers ever mentioning a circle definition of the tangent function.

The geometric definition of the tangent function, which predates the triangle definition, is the length of a segment tangent to the unit circle. The tangent really is a tangent! Just as for sine and cosine, this one-variable definition helps develop intuition. Here is the definition, followed by an applet to help you get a feel for it:


Trigonometry Yoga

Which is bigger, the sine of 40\(^{\circ}\) or the sine of 50\(^{\circ}\)?

This is a great trigonometry assessment question.   Unfortunately, virtually none of my college students who haven’t used trig for a while can answer it (without a calculator).   For a lot of students, trig is one of those subjects that just didn’t stick very well.

So, we start trig functions all over, but this time with circle definitions and some “trigonometry yoga” (no trademark; I made up the name when I was thinking of a title for this post).