We now turn to a little-taught and perhaps controversial inverse function: arcsecant. As we’ve seen throughout this series, the domain of this inverse function must be chosen so that the graph of satisfies the horizontal line test. It turns out that the choice of domain has surprising consequences that are almost unforeseeable using only the tools of Precalculus.

The standard definition of uses the interval — or, more precisely, to avoid the vertical asymptote at . This portion of the graph of satisfies the horizontal line test and, conveniently, matches almost perfectly the domain of . This is perhaps not surprising since, when both are defined, and are reciprocals.

Since this range of matches that of , we have the convenient identity

To see why this works, let’s examine the right triangle below. Notice that

.

Also,

.

This argument provides the justification for — that is, for — but it still works for and .

So this seems like the most natural definition in the world for . Unfortunately, there are consequences for this choice in calculus, as we’ll see in tomorrow’s post.

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