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Chapter 4: Problem 51

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

Short Answer

Expert verified
The solutions to the equation in the given interval are \(x = -5\pi/3, -2\pi/3, \pi/3, 4\pi/3\).

Step by step solution

01

- Understanding Cotangent Graph

The cotangent function, represented as \(cot x\), is a periodic function with a period of \(\pi\). This means it repeats every \(\pi\) units. The cotangent function is undefined at \(x = n\pi\), where \(n\) is an integer, because at these points the cotangent function approaches infinity. Hence, in solving this equation, consider the fact that \(\cot x\) has points of discontinuity at these multiples of \(\pi\).
02

- Finding Reference Angle

The given equation is \(\cot x = -\sqrt{3}/3\). To make things easier, the value of \(-\sqrt{3}/3\) can be compared to standard values of trigonometric functions to find the reference angle. The cotangent of \(\pi/6\) is \(\sqrt{3}\), thus cotangent of \(-\pi/6\) is \(-\sqrt{3}\). But -\sqrt{3}/3 is not equal to \(-\sqrt{3}\), so the actual angle that gives the cotangent value of \(-\sqrt{3}/3\) would be \(-\pi/3\).
03

- Finding Solutions

Since cotangent has a period of \(\pi\), the general rule for the solutions of the cotangent function is \(x = -\pi/3 + n\pi\), where \(n\) is an integer. This is based on the reference angle and the period of the cotangent function. So continuing, the solutions for \(x\) in the required interval \(-2\pi, 2\pi\) are obtained by substituting integer values of \(n\) from -2 to 2 in the equation \(x = -\pi/3 + n\pi\). The solutions for these values of n that lie in the given interval are \(x = -5\pi/3, -2\pi/3, \pi/3, 4\pi/3\).

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Most popular questions from this chapter

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