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Chapter 4: Problem 55

Use a graph to solve the equation on the interval \(-2 \pi, 2 \pi\). $$\csc x=\sqrt{2}$$

Short Answer

Expert verified
The solutions to the equation are \(x=\pm \pi/4, \pm 3\pi/4\) within the specified interval \(-2\pi \leq x \leq 2\pi\).

Step by step solution

01

Understanding Cosecant Function

The cosecant function, \(\csc x\), is the reciprocal of the sine function. Thus, \(\csc x = 1/ \sin x\). Since sine is periodic with a period of \(2\pi\), cosecant is also periodic with the same period, but it is undefined at \(x=n\pi\), where \(n\) is an integer because the sine function is zero at these points.
02

Converting the Equation

The given equation is \(\csc x=\sqrt{2}\). This can be converted into a sine equation since \(\csc x = 1/ \sin x\). Therefore, \(\csc x = \sqrt{2}\) is equivalent to \(1/ \sin x = \sqrt{2}\) or \(\sin x = 1/ \sqrt{2}\).
03

Solving the Equation

To find the solutions of this equation within the given interval \(-2\pi \leq x \leq 2\pi\), note that sine function attains the value \(1/ \sqrt{2}\) at \(x=\pi/4\) and \(x=3\pi/4\) in the interval \([0, \pi]\), and the function is periodic with a period of \(2\pi\). Moreover, sine function is symmetric with respect to the origin. Hence, the solutions to the equation \(\sin x = 1/ \sqrt{2}\) in the given interval are \(x=\pm \pi/4, \pm 3\pi/4\).
04

Double-Check with Graph

It can be helpful to sketch a graph of the function \(\sin x\) and the horizontal line \(y = 1/ \sqrt{2}\) to visually confirm the solutions obtained in Step 3. The x-coordinates of the points of intersection between the graph of \(\sin x\) and the horizontal line are the solutions to the given equation.

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Most popular questions from this chapter

Sketch a graph of the function and compare the graph of \(g\) with the graph of \(f(x)=\arcsin x\). $$g(x)=\arcsin (x-1)$$

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\) (a) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\tan x$$

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