My Favorite One-Liners: Part 76

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Here’s a problem that might arise in trigonometry:

Compute .

To begin, we observe that , so that

.

We then remember that is a periodic function with period . This means that we can add or subtract any multiple of to the angle, and the result of the function doesn’t change. In particular, is a multiple of , so that

.

Said another way, corresponds to complete rotations, and the value of cosine doesn’t change with a complete rotation. So it’s OK to just throw away any even multiple of when computing the sine or cosine of a very large angle. I then tell my class:

In mathematics, there’s a technical term for this idea; it’s called throwing.