##### The Domain and Range of the Inverse Trigonometric Functions

Recall that if the function\(f\) is the set of ordered pairs \(\{(a,b)\}\text{,}\) then the inverse function, if it exists, is the set of ordered pairs \(\{(b,a)\}\text{.}\) Now suppose that \(y=f(x)\text{.}\) Because a function cannot have two ordered pairs with the same \(x\)-coordinate, \(f\) will have an inverse if and only if no two ordered pairs have the same \(y\)-coordinate. Such a function is called a one-to-one function and function is invertible if and only if it is one-to-one.

The concept of one-to-one is not a relative term. That is, a function is either one-to-one or it's not. That said, if it were a relative term then the six basic trigonometric functions would all be not one-to-one in the extreme. If a number is in the range of a basic trigonometric function, then there are an endless number of ordered pairs in the set with a \(y\)-coordinate equal to that value. Equivalently, there are an endless number of of points on a graph of the function with a \(y\)-coordinate equal to that number.

Since none of the six basic trigonometric functions is one-to-one, none of them have an inverse function unless we restrict the domain of the function. When choosing the domain restrictions, we want to choose domains that cover the entire range of the function. For example, we don't want to restrict the domain to \(\left(0,\frac{\pi}{2}\right)\text{,}\) because that would limit the ranges to subsets of the positive real numbers.

Ideally the domain restrictions would be the same for all six basic trigonometric functions, but this simply is not possible. While its true that all six of the functions are positive for angles that terminate in Quadrant I, there is no Quadrant where all six trigonometric functions are negative. Also, there are discontinuities in the domains of the tangent, cotangent, secant, and cosecant functions while the same is not true for the sine and cosine functions.

Graphs of the six basic trigonometric functions are shown in FigureĀ 14.11.1-FigureĀ 14.11.6. The domain restriction is stated for each function and the resultant restricted portion of the function curve is highlighted. Notice that each restriction still covers the entire range of the function. Notice as well that over each restricted domain the function is one-to-one.

Recall that the domain and range of a function, \(f\text{,}\) are, respectively the range and domain of of its inverse function (assuming that an inverse function exists). The restricted domains and the ranges of the six basic trigonometric functions and the domains and ranges of their inverse functions are stated below (and graphically implied above). It's important to remember that these domain restrictions are only applicable in this context of inverse functions.

The domain of the function \(y=\sin(t)\) is restricted to \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\) and the range of the function \(y=\sin(t)\) is \([-1,1]\text{.}\) The domain of the function \(y=\sin^{-1}(t)\) (read either as "y equals the inverse sine of t" or as "y equals the arcsine of t") is \([-1,1]\) and the range of the function \(y=\sin^{-1}(t)\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\text{.}\)

The domain of the function \(y=\cos(t)\) is restricted to \([0,\pi]\) and the range of the function \(y=\cos(t)\) is \([-1,1]\text{.}\) The domain of the function \(y=\cos^{-1}(t)\) is \([-1,1]\) and the range of the function \(y=\cos^{-1}(t)\) is \([0,\pi]\text{.}\)

The domain of the function \(y=\tan(t)\) is restricted to \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\) and the range of the function \(y=\tan(t)\) is \((-\infty,\infty)\text{.}\) The domain of the function \(y=\tan^{-1}(t)\) is \((-\infty,\infty)\) and the range of the function \(y=\tan^{-1}(t)\) is \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\text{.}\)

The domain of the function \(y=\cot(t)\) is restricted to \((0,\pi)\) and the range of the function \(y=\cot(t)\) is \((-\infty,\infty)\text{.}\) The domain of the function \(y=\cot^{-1}(t)\) is \((-\infty,\infty)\) and the range of the function \(y=\cot^{-1}(t)\) is \((0,\pi)\text{.}\)

The domain of the function \(y=\sec(t)\) is restricted to \(\left[0,\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right]\) and the range of the function \(y=\sec(t)\) is \((-\infty,1] \cup [1,\infty)\text{.}\) The domain of the function \(y=\sec^{-1}(t)\) is \((-\infty,1] \cup [1,\infty)\) and the range of the function \(y=\sec^{-1}(t)\) is \(\left[0,\frac{\pi}{2}\right) \cup \left(\frac{\pi}{2},\pi\right]\text{.}\)

The domain of the function \(y=\csc(t)\) is restricted to \(\left[-\frac{\pi}{2},0\right) \cup \left(0,\frac{\pi}{2}\right]\) and the range of the function \(y=\csc(t)\) is \((-\infty,1] \cup [1,\infty)\text{.}\) The domain of the function \(y=\csc^{-1}(t)\) is \((-\infty,1] \cup [1,\infty)\) and the range of the function \(y=\csc^{-1}(t)\) is \(\left[-\frac{\pi}{2},0\right) \cup \left(0,\frac{\pi}{2}\right]\text{.}\)