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Chapter 4: Problem 80

In Exercises \(61-86,\) use reference angles to find the exact value of each expression. Do not use a calculator. $$\cot \frac{13 \pi}{3}$$

Short Answer

Expert verified
The exact value of \( \cot (\frac{13\pi}{3}) \) is \( \frac{\sqrt{3}}{3} \).

Step by step solution

01

Determine the Coterminal Angle

Firstly, in order to simplify the process, find a coterminal angle that lies between \(0\) and \(2\pi\). This can be achieved by subtracting \(2\pi\) until you get the angle in that interval. In this case, cotangles of \(\frac{13\pi}{3}\) would be \(\frac{13\pi}{3} - \frac{12\pi}{3}\), resulting in \(\frac{\pi}{3}\), which falls under that interval.
02

Find the reference angle

The reference angle of any angle in standard position is the acute angle formed by the terminal side of the angle and the x-axis. Here, since \( \frac{\pi}{3} \) lies in the first quadrant, the reference angle is the angle itself, meaning \(\frac{\pi}{3}\).
03

Use the Reference Angle to find the Cotangent

The cotangent of an angle in standard position equals the cosine of the reference angle divided by the sine of the reference angle. In the unit circle, the sine of \( \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{2} \) and the cosine of \( \frac{\pi}{3} \) is \( \frac{1}{2} \). Therefore, the cotangent of \( \frac{13\pi}{3} \) equals \( \frac{cos(\frac{\pi}{3})}{sin(\frac{\pi}{3})} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\). Finally, rationalize the denominator to get \( \frac{\sqrt{3}}{3} \).

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