Chapter 4.6, Problem 55E

To determine

To find: the solutions of the equation in the interval $\left[-2\pi ,2\pi \right]$ .

Answer to Problem 55E

  $\csc x=\sqrt{2}\text{a}tx=-\left(\frac{7\pi }{4}\right),x=-\left(\frac{5\pi }{4}\right),x=\left(\frac{\pi }{4}\right)$ and $x=\left(\frac{3\pi }{4}\right)$ .

Explanation of Solution

Given information:

Given function is

  $\csc x=\sqrt{2}$ .

Calculation:

Let us consider the points to solve the equation $\csc x=\sqrt{2}$ .

Let us, first plot the graph for $f\left(x\right)=\csc x$

Now, in the above graph, look for all the points on the graph where the value of $\csc x$ is $\sqrt{2}$ .

From the above graph, it can be seen that the value of $\csc x$ equal to $\sqrt{2}$ at

  $x=\left(\frac{7\pi }{4}\right),-\left(\frac{5\pi }{4}\right),\left(\frac{\pi }{4}\right)$ and $\left(\frac{3\pi }{4}\right)$

Therefore,

  $\csc x=\sqrt{2}\text{a}tx=-\left(\frac{7\pi }{4}\right),x=-\left(\frac{5\pi }{4}\right),x=\left(\frac{\pi }{4}\right)$ and $x=\left(\frac{3\pi }{4}\right)$ .