Trigonometry/Inverse Trigonometric Functions

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The Inverse Functions, Restrictions, and Notation

While it might seem that inverse trigonometric functions should be relatively self defining, some caution is necessary to get an inverse function since the trigonometric functions are not one-to-one. To deal with this issue, some texts have adopted the convention of allowing ${\sin }^{-1}x$, ${\cos }^{-1}x$, and ${\tan }^{-1}x$ (all with lower-case initial letters) to indicate the inverse relations for the trigonometric functions and defining new functions $\text{Sin}x$, $\text{Cos}x$, and $\text{Tan}x$ (all with initial capitals) to equal the original functions but with restricted domain, thus creating one-to-one functions with the inverses ${\text{Sin}}^{-1}x$, ${\text{Cos}}^{-1}x$, and ${\text{Tan}}^{-1}x$. For clarity, we will use this convention. Another common notation used for the inverse functions is the "arcfunction" notation: ${\text{Sin}}^{-1}x=\arcsin x$, ${\text{Cos}}^{-1}x=\arccos x$, and ${\text{Tan}}^{-1}x=\arctan x$ (the arcfunctions are sometimes also capitalized to distinguish the inverse functions from the inverse relations). The arcfunctions may be so named because of the relationship between radian measure of angles and arclength--the arcfunctions yeild arc lengths on a unit circle.

The restrictions necessary to allow the inverses to be functions are standard: ${\text{Sin}}^{-1}x$ has range $[-\begin{array}{c}\frac{\pi }{2}\end{array},\begin{array}{c}\frac{\pi }{2}\end{array}]$; ${\text{Cos}}^{-1}x$ has range $[0,\pi ]$; and ${\text{Tan}}^{-1}x$ has range $(-\begin{array}{c}\frac{\pi }{2}\end{array},\begin{array}{c}\frac{\pi }{2}\end{array})$ (these restricted ranges for the inverses are the restricted domains of the capital-letter trigonometric functions). For each inverse function, the restricted range includes first-quadrant angles as well as an adjacent quadrant that completes the domain of the inverse function and maintains the range as a single interval.

It is important to note that because of the restricted ranges, the inverse trigonometric functions do not necessarily behave as one might expect an inverse function to behave. While ${\text{Sin}}^{-1}(\sin \left(\begin{array}{c}\frac{\pi }{6}\end{array}\right))={\text{Sin}}^{-1}\left(\begin{array}{c}\frac{1}{2}\end{array}\right)=\begin{array}{c}\frac{\pi }{6}\end{array}$ (following the expected ${\text{Sin}}^{-1}(\sin x)=x$), ${\text{Sin}}^{-1}(\sin \left(\begin{array}{c}\frac{5\pi }{6}\end{array}\right))={\text{Sin}}^{-1}\left(\begin{array}{c}\frac{1}{2}\end{array}\right)=\begin{array}{c}\frac{\pi }{6}\end{array}$! For the inverse trigonometric functions, $f^{-1}\circ f(x)=x$ only when $x$ is in the range of the inverse function. The other direction, however, is less tricky: $f\circ f^{-1}(x)=x$ for all $x$ to which we can apply the inverse function.

The Inverse Relations

For the sake of completeness, here are definitions of the inverse trigonometric relations based on the inverse trigonometric functions:

• ${\sin }^{-1}x=\{{\text{Sin}}^{-1}x+2\pi n,n\in \mathbb{Z}\}\cup \{\pi -{\text{Sin}}^{-1}x+2\pi n,n\in \mathbb{Z}\}$ (the sine function has period $2\pi$, but within any given period may have two solutions and $\sin x=\sin (\pi -x)$)
• ${\cos }^{-1}x=\{\pm {\text{Cos}}^{-1}x+2\pi n,n\in \mathbb{Z}\}$ (the cosine function has period $2\pi$, but within any given period may have two solutions and cosine is even--$\cos x=\cos (-x)$)
• ${\tan }^{-1}x=\{{\text{Tan}}^{-1}x+\pi n,n\in \mathbb{Z}\}$ (the tangent function has period $\pi$ and is one-to-one within any given period)

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