5.3 Inverse Trigonometric Functions

We have briefly mentioned the inverse trigonometric functions before, for example in Section 1.3 when we discussed how to use the 1pt \(\sin^{-1}\), 1pt \(\cos^{-1}\), and 1pt \(\tan^{-1}\) buttons on a calculator to find an angle that has a certain trigonometric function value. We will now define those inverse functions and determine their graphs.

Recall that a function is a rule that assigns a single object \(y\) from one set (the range) to each object \(x\) from another set (the domain). We can write that rule as \(y = f(x)\), where \(f\) is the function (see Figure 5.3.1). There is a simple vertical rule for determining whether a rule \(y=f(x)\) is a function: \(f\) is a function if and only if every vertical line intersects the graph of \(y=f(x)\) in the \(xy\)-coordinate plane at most once (see Figure 5.3.2).

Recall that a function \(f\) is one-to-one (often written as \(1-1\)) if it assigns distinct values of \(y\) to distinct values of \(x\). In other words, if \(x_1 \ne x_2\) then \(f(x_1 ) \ne f(x_2 )\). Equivalently, \(f\) is one-to-one if \(f(x_1 ) = f(x_2 )\) implies \(x_1 = x_2\). There is a simple horizontal rule for determining whether a function \(y=f(x)\) is one-to-one: \(f\) is one-to-one if and only if every horizontal line intersects the graph of \(y=f(x)\) in the \(xy\)-coordinate plane at most once (see Figure 5.3.3).

If a function \(f\) is one-to-one on its domain, then \(f\) has an inverse function, denoted by \(f^{-1}\), such that \(y=f(x)\) if and only if \(f^{-1}(y) = x\). The domain of \(f^{-1}\) is the range of \(f\).

The basic idea is that \(f^{-1}\) “undoes” what \(f\) does, and vice versa. In other words,

We know from their graphs that none of the trigonometric functions are one-to-one over their entire domains. However, we can restrict those functions to subsets of their domains where they are one-to-one. For example, \(y=\sin\;x\) is one-to-one over the interval \(\left[ -\frac{\pi}{2},\frac{\pi}{2} \right]\), as we see in the graph below:

For \(-\frac{\pi}{2} \le x \le \frac{\pi}{2}\) we have \(-1 \le \sin\;x \le 1\), so we can define the inverse sine function \(y=\sin^{-1} x\) (sometimes called the arc sine and denoted by \(y=\arcsin\;x\)) whose domain is the interval \(\ival{-1}{1}\) and whose range is the interval \(\left[ -\frac{\pi}{2},\frac{\pi}{2} \right]\). In other words:

Example 5.14 illustrates an important point: \(\sin^{-1} x\) should always be a number between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). If you get a number outside that range, then you made a mistake somewhere. This why in Example 1.27 in Section 1.5 we got \(\sin^{-1}(-0.682) = -43\Degrees\) when using the 1pt \(\sin^{-1}\) button on a calculator. Instead of an angle between \(0\Degrees\) and \(360\Degrees\) (i.e. \(0\) to \(2\pi\) radians) we got an angle between \(-90\Degrees\) and \(90\Degrees\) (i.e. \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) radians).

In general, the graph of an inverse function \(f^{-1}\) is the reflection of the graph of \(f\) around the line \(y=x\). The graph of \(y=\sin^{-1} x\) is shown in Figure 5.3.5. Notice the symmetry about the line \(y=x\) with the graph of \(y=\sin\;x\).

The inverse cosine function \(y=\cos^{-1} x\) (sometimes called the arc cosine and denoted by \(y=\arccos\;x\)) can be determined in a similar fashion. The function \(y=\cos\;x\) is one-to-one over the interval \(\ival{0}{\pi}\), as we see in the graph below:

Thus, \(y=\cos^{-1} x\) is a function whose domain is the interval \(\ival{-1}{1}\) and whose range is the interval \(\ival{0}{\pi}\). In other words:

The graph of \(y=\cos^{-1} x\) is shown below in Figure 5.3.7. Notice the symmetry about the line \(y=x\) with the graph of \(y=\cos\;x\).

Examples 5.14 and 5.16 may be confusing, since they seem to violate the general rule for inverse functions that \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f\). But that rule only applies when the function \(f\) is one-to-one over its entire domain. We had to restrict the sine and cosine functions to very small subsets of their entire domains in order for those functions to be one-to-one. That general rule, therefore, only holds for \(x\) in those small subsets in the case of the inverse sine and inverse cosine. The inverse tangent function \(y=\tan^{-1} x\) (sometimes called the arc tangent and denoted by \(y=\arctan\;x\)) can be determined similarly. The function \(y=\tan\;x\) is one-to-one over the interval \(\left( -\frac{\pi}{2},\frac{\pi}{2} \right)\), as we see in Figure 5.3.8:

The graph of \(y=\tan^{-1} x\) is shown below in Figure 5.3.9. Notice that the vertical asymptotes for \(y=\tan\;x\) become horizontal asymptotes for \(y=\tan^{-1} x\). Note also the symmetry about the line \(y=x\) with the graph of \(y=\tan\;x\).

Thus, \(y=\tan^{-1} x\) is a function whose domain is the set of all real numbers and whose range is the interval \(\left( -\frac{\pi}{2},\frac{\pi}{2} \right)\). In other words:

The inverse functions for cotangent, cosecant, and secant can be determined by looking at their graphs. For example, the function \(y=\cot\;x\) is one-to-one in the interval \((0,\pi)\), where it has a range equal to the set of all real numbers. Thus, the inverse cotangent \(y=\cot^{-1} x\) is a function whose domain is the set of all real numbers and whose range is the interval \((0,\pi)\). In other words:

The graph of \(y=\cot^{-1} x\) is shown below in Figure 5.3.11.

Similarly, it can be shown that the inverse cosecant \(y=\csc^{-1} x\) is a function whose domain is \(\abs{x} \ge 1\) and whose range is \(-\frac{\pi}{2} \le y \le \frac{\pi}{2}\), \(y \ne 0\). Likewise, the inverse secant \(y=\sec^{-1} x\) is a function whose domain is \(\abs{x} \ge 1\) and whose range is \(0 \le y \le \pi\), \(y \ne \frac{\pi}{2}\).

It is also common to call \(\cot^{-1} x\), \(\csc^{-1} x\), and \(\sec^{-1} x\) the arc cotangent, arc cosecant, and arc secant, respectively, of \(x\). The graphs of \(y=\csc^{-1} x\) and \(y=\sec^{-1} x\) are shown in Figure 5.3.12: