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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, sec\(^{-1}(x)\) can be defined on the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi]\). Its graph is obtained by reflecting the graph of the restricted secant function over the line y = x. The upwards 'U' shaped portion of the secant function turns into a rightward opening 'U' and the downwards 'U' into a leftwards opening 'U' for the inverse secant function.

Step by step solution

01

- Understanding the secant function

We begin by understanding the secant function. The secant function, sec(x), is the reciprocal of the cosine function, meaning sec(x) = 1/cos(x). Notice that sec(x) is undefined when cos(x) = 0.
02

- Restricting the domain

To define an inverse, sec\(^{-1}(x)\), we have to restrict the domain of the secant function so it passes the horizontal line test and becomes a one-to-one function. With the provided intervals, \([0, \pi / 2)\) and \((\pi / 2, \pi]\), we exclude the values where the secant function has vertical asymptotes and is undefined.
03

- Sketching the restricted secant function

On graph paper, sketch the secant function on the restricted intervals. Notice it forms a U shape on the interval \([0, \pi/2)\) and an upside-down U on the interval \((\pi/2, \pi]\). Vertical asymptotes occur at x = \(\pi / 2\).
04

- Sketching the inverse function

The inverse function is the reflection of the original function over the line y = x. So, sketch a line y = x and reflect the restricted secant graph over the line. The upward-opening 'U' shape becomes a rightward-opening 'U' for the inverse secant function, and the downward 'U' becomes a leftward-opening 'U'.

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Most popular questions from this chapter

Write a short paper explaining to a classmate how to evaluate the six trigonometric functions of any angle \(\theta\) in standard position. Include an explanation of reference angles and how to use them, the signs of the functions in each of the four quadrants, and the trigonometric values of common angles. Be sure to include figures or diagrams in your paper.

\(\quad\) A point on the end of a tuning fork moves in simple harmonic motion described by \(d=a \sin \omega t .\) Find \(\omega\) given that the tuning fork for middle C has a frequency of 264 vibrations per second.

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Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse trigonometric function.
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