Answered: Consider the function f(x) = tan(sin…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter1: Trigonometry

Section: Chapter Questions

Problem 15T: In Exercises 14 and 15, use a graphing utility to graph the function. If the function is periodic,...

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Question

Consider the function f(x) = tan(sin x). (a) Use a graphing utility to graph the function. (b) Identify any symmetry of the graph. (c) Is the function periodic? If so, what is the period? (d) Identify any extrema on (−1, 1). (e) Use a graphing utility to determine the concavity of the graph on (0, 1).

Expert Solution

Step 1

The given function $f (x) = \tan (\sin x)$ .

We have to sketch the graph of the function and find its symmetry, period, extrema on $(- 1, 1)$ and concavity on $(0, 1)$ .

Step 2

$f (x) = \tan (\sin x)$

Use Desmos graphing tool to graph the function.

Graph of the function:

This is the required graph.

Step 3

$f (x) = \tan (\sin x)$

$\begin{array}{rcl} f (- x) & = & \tan (\sin (- x)) \\ = & \tan (- \sin x) \\ = & - \tan (\sin x) \\ f (- x) & = & - f (x) \end{array}$

The function is odd function.

Thus, the graph is symmetry about origin.

The function is periodic.

From the graph, one complete cycle between $- \frac{π}{2}$ to $\frac{3 π}{2}$ .

Period,

$\begin{array}{rcl} P & = & \frac{3 π}{2} - (- \frac{π}{2}) \\ = & \frac{3 π}{2} + \frac{π}{2} \\ = & \frac{4 π}{2} \\ P & = & 2 π \end{array}$

Therefore, the function is periodic and its period is $2 π$ .

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