My Favorite One-Liners: Part 63

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.

I’ll use today’s one-liner to explain why mathematicians settled on a particular convention that could have been chosen differently. For example, let’s consider the definition of  $y = \sin^{-1} x$ by first looking at the graph of $f(x) = \sin x$.

Of course, we can’t find an inverse for this function; colloquially, the graph of $f$ fails the horizontal line test. More precisely, there exist two numbers $x_1$ and $x_2$ so that $x_1 \ne x_2$ but $f(x_1) = f(x_2)$. Indeed, there are infinitely many such pairs.

So how will we find the inverse of $f$? Well, we can’t. But we can do something almost as good: we can define a new function $g$ that’s going look an awful lot like $f$. We will restrict the domain of this new function $g$ so that $g$ satisfies the horizontal line test.

For the sine function, there are plenty of good options from which to choose. Indeed, here are four legitimate options just using the two periods of the sine function shown above. The fourth option is unorthodox, but it nevertheless satisfies the horizontal line test (as long as we’re careful with $\pm 2\pi$.

So which of these options should we choose? Historically, mathematicians have settled for the interval $[-\pi/2, \pi/2]$.

So, I’ll ask my students, why have mathematicians chosen this interval? That I can answer with one word: tradition.

For further reading, see my series on inverse functions.