Inverse Trig Functions

Setting up the Inversion

Just before we begin, we need to remember that the $textcolor{blue}{sin}$, $textcolor{limegreen}{cos}$ and $textcolor{red}{tan}$ graphs are not one-to-one functions, they are many-to-one functions. We have a set of $x$ values which give back the same result.

For example, we have $textcolor{blue}{sin} 45 = textcolor{blue}{sin} 135 = textcolor{blue}{sin} 405 = textcolor{blue}{sin} 495 = …$

A function can only have an inverse if it is one-to-one. As a result, we’ll need to restrict the domain (the range of values of $x$) of each original trig graph to be one-to-one:

• $textcolor{blue}{sin x}$ is restricted to
  • $dfrac{-pi}{2} leq x leq dfrac{pi}{2}$
  • $-1 leq textcolor{blue}{sin x} leq 1$
• $textcolor{limegreen}{cos x}$ is restricted to
  • $0 leq x leq pi$
  • $-1 leq textcolor{limegreen}{cos x} leq 1$
• $textcolor{red}{tan x}$ is restricted to
  • $dfrac{-pi}{2} < x < dfrac{pi}{2}$
  • The range of $textcolor{red}{tan x}$ is unrestricted

Here’s the $textcolor{blue}{sin x}$ graph, along with $textcolor{purple}{sin ^{-1}x}$…

… and now the $textcolor{limegreen}{cos x}$ graph, along with $textcolor{purple}{cos ^{-1}x}$…

… and the $textcolor{red}{tan x}$ graph, along with $textcolor{purple}{tan ^{-1}x}$.

Inverse Representation in Graphical Form

Actually, you might have noticed that the inverse function is the original function reflected in the line $y = x$.

This is true for any inverse function, but this is probably a good time to mention it anyway.