8.3 Inverse trigonometric functions  (Page 2/15)

On these restricted domains, we can define the inverse trigonometric functions .

• The inverse sine function     $y={\sin }^{-1}x$ means $x=\sin y.$ The inverse sine function is sometimes called the arcsine    function, and notated $\arcsin x.$
  $$y={\sin }^{-1}x\text{has domain}[\mathrm{-1},1]\text{and range}\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$$
• The inverse cosine function     $y={\cos }^{-1}x$ means $x=\cos y.$ The inverse cosine function is sometimes called the arccosine    function, and notated $\arccos x.$
  $y={\cos }^{-1}x\text{has domain}[\mathrm{-1},1]\text{and range}[0,\pi ]$
• The inverse tangent function     $y={\tan }^{-1}x$ means $x=\tan y.$ The inverse tangent function is sometimes called the arctangent    function, and notated $\arctan x.$
  $y={\tan }^{-1}x\text{has domain}(\mathrm{-\infty },\infty )\text{and range}\left(-\frac{\pi }{2},\frac{\pi }{2}\right)$

The graphs of the inverse functions are shown in [link] , [link] , and [link] . Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that ${\sin }^{-1}x$ has domain $\left[\mathrm{-1},1\right]$ and range $\left[-\frac{\pi }{2},\frac{\pi }{2}\right],$ ${\cos }^{-1}x$ has domain $\left[\mathrm{-1}\mathrm{,1}\right]$ and range $[0,\pi ],$ and ${\tan }^{-1}x$ has domain of all real numbers and range $\left(-\frac{\pi }{2},\frac{\pi }{2}\right).$ To find the domain    and range    of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line $y=x.$

Finding the exact value of expressions involving the inverse sine, cosine, and tangent functions

Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse functions when we are using the special angles, specifically $\frac{\pi }{6}$ (30°), $\frac{\pi }{4}$ (45°), and $\frac{\pi }{3}$ (60°), and their reflections into other quadrants.