Agreed upon domain of inverse trig functions

Why do we choose $[-\pi/2,\pi/2]$ as the domain restriction for $\sin(x)$ when determining $\arcsin(x)$, clearly other restrictions are possible, which also yield a 1-1 function (like $[\pi/2,3\pi/2]$). No pre-calculus book I have seen goes beyond saying that this choice is completely arbitrary. However, I have noticed that every domain restriction for inverse trig functions contains the whole first quadrant in its entirety (with the exception of x values where there are vertical asymptopes). I think these domain restrictions we choose are the only possible choices to get a 1-1 function on a domain that includes "effectively" the whole first quadrant. Is this the reason why we choose these domain restrictions? Are there other reasons why these domain restrictions are more natural than others that would yield a 1-1 function?

Solution 1:

It's nice to have a domain that includes $0$. The domain of a trig function is the range of the corresponding inverse function, and it's nice to have $0$ in the range.

Think of applications. If I'm building something, and I want to set two pieces at some angle, I want an angle between $0$ and $\pi$, because that's what I can measure with my tools. That's one reason that it's nice to use intervals either centered on $0$, or with $0$ as an endpoint.

Also, in calculus, we like to represent functions as power series, and these things are a lot easier to write down if we can center them at $x=0$. Of course, this isn't an explanation that would have much place in one of those Precalculus books that you mention, but it would seem strange to define the ranges of inverse trig functions to be one thing in Precal and something else in Calculus.

There may be other reasons, but I think these two are compelling.