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 \cos \displaystyle \frac{2017\pi}{6}.

To begin, we observe that \displaystyle \frac{2017}{6} = 336 + \displaystyle \frac{1}{6}, so that

\cos \displaystyle \frac{2017\pi}{6} = \cos \left( \displaystyle 336\pi + \frac{\pi}{6} \right).

We then remember that \cos \theta is a periodic function with period 2\pi. This means that we can add or subtract any multiple of 2\pi to the angle, and the result of the function doesn’t change. In particular, -336\pi is a multiple of 2 \pi, so that

\cos \displaystyle \frac{2017\pi}{6} = \cos \left( \displaystyle 336\pi + \frac{\pi}{6} \right)

= \cos \left( \displaystyle 336\pi + \frac{\pi}{6} - 336\pi \right)

= \cos \displaystyle \frac{\pi}{6}

= \displaystyle \frac{\sqrt{3}}{2}.

Said another way, 336\pi corresponds to 336/2 = 168 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 \pi 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 \pi throwing.

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