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Chapter 4: Problem 116

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse trigonometric function.

Short Answer

Expert verified
The inverse secant function, defined over the restricted domain of \(sec(x)\) i.e. \([0, \pi / 2)\) and \((\pi / 2, \pi]\), yields an increasing function in these intervals when graphed. The graph of the inverse function is a reflection of original function about the line \(y = x\).

Step by step solution

01

Restrict the Domain of the Secant Function

We are given the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi]\). Thus, we restrict the domain of the secant function \(sec(x)\) to these intervals. We have to exclude \(\pi / 2\) because \(sec(\pi / 2)\) is undefined.
02

Define the Inverse Secant Function

The inverse secant function, denoted as \(sec^{-1}(x)\) or \(arcsec(x)\), is defined as a function that returns an angle whose secant is \(x\). It un-does what the secant function does. Remember that the inverse secant function is only defined over the restricted domain established in step 1.
03

Sketch the Graph of the Inverse Trigonometric Function

Now, to sketch the graph of the inverse secant function, recall that the graph of an inverse function is a reflection of the original function across the line \(y = x\). Start by plotting key points of the secant function and reflect them over the line \(y = x\). The graph of \(sec^{-1}(x)\) should be increasing in the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi]\), which corresponds to the restricted domain of the secant function. Note that the range of \(sec^{-1}(x)\) is \([0, \pi)\) excluding \(\pi / 2\), and it should not possess any horizontal asymptotes in the given intervals.

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Most popular questions from this chapter

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window to verify that the two functions are equal. Explain why they are equal. Identify any asymptotes of the graphs. \(f(x)=\tan \left(\arccos \frac{x}{2}\right), \quad g(x)=\frac{\sqrt{4-x^{2}}}{x}\)

Find the exact value of the expression. (Hint: Sketch a right triangle.) \(\cot \left(\arctan \frac{5}{8}\right)\)

Evaluate the expression without using a calculator. \(\arctan \sqrt{3}\)

Evaluate the expression without using a calculator. \(\arctan 1\)

Fill in the blank. If not possible, state the reason. As \(x \rightarrow-1^{+},\) the value of \(\arccos x \rightarrow\) ___.
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