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Chapter 4: Problem 56

Use a graph to solve the equation on the interval \(-2 \pi, 2 \pi\). $$\csc x=-\frac{2 \sqrt{3}}{3}$$

Short Answer

Expert verified
The solutions to the equation \(\csc x = -\frac{2 \sqrt{3}}{3}\) on the interval \(-2 \pi, 2 \pi\) are \(x \approx -\frac{5 \pi}{3}, -\frac{2 \pi}{3}, \frac{\pi}{3}, \frac{4 \pi}{3}\).

Step by step solution

01

Understand the function \(\csc x\)

The function \(\csc x\) is the reciprocal of the sine function, meaning that \(\csc x = \frac{1}{\sin x}\). This relationship also means that wherever sine equals zero, cosecant is undefined, and vice-versa.
02

Determine graphical representation of \(y = \csc x\)

Draw the \(\csc x\) graph over the interval \(-2 \pi, 2 \pi\). Remember that \(\csc x\) has vertical asymptotes wherever the sine function is zero, which happens at integral multiples of \(\pi\).
03

Understand the line \(y = -\frac{2 \sqrt{3}}{3}\)

Now draw the horizontal line of \(y = -\frac{2 \sqrt{3}}{3}\) on the same plot. This line represents the right side of the equation.
04

Find the intersection

Identify the points where the \(\csc x\) graph and the line \(y = -\frac{2 \sqrt{3}}{3}\) intersect. These are the x values that satisfy the given equation on the interval \(-2 \pi, 2 \pi\). The solution set is the set of all these x values. For the given equation, these points occur at \(x \approx -\frac{5 \pi}{3}, -\frac{2 \pi}{3}, \frac{\pi}{3}, \frac{4 \pi}{3}\)

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