Question

use reference angles to find the exact value of the following expression. do not use a calculator. cot $\frac{29pi}{6}$ select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. cot $\frac{29pi}{6}$ = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.) b. the answer is undefined.

Explanation:

Step1: Find a coterminal angle

Subtract \(4\pi\) (or \(\frac{24\pi}{6}\)) from \(\frac{29\pi}{6}\). So \(\frac{29\pi}{6}-4\pi=\frac{29\pi}{6}-\frac{24\pi}{6}=\frac{5\pi}{6}\).

Step2: Determine the reference angle

The angle \(\frac{5\pi}{6}\) is in the second - quadrant. The reference angle \(\theta'\) for \(\theta=\frac{5\pi}{6}\) is \(\pi - \frac{5\pi}{6}=\frac{\pi}{6}\).

Step3: Evaluate the cotangent of the reference angle

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), and for \(\theta = \frac{\pi}{6}\), \(\sin\frac{\pi}{6}=\frac{1}{2}\), \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\), so \(\cot\frac{\pi}{6}=\frac{\cos\frac{\pi}{6}}{\sin\frac{\pi}{6}}=\sqrt{3}\).

Step4: Determine the sign of the cotangent in the second - quadrant

In the second - quadrant, \(\sin\theta> 0\) and \(\cos\theta<0\), so \(\cot\theta=\frac{\cos\theta}{\sin\theta}<0\).
So \(\cot\frac{5\pi}{6}=-\sqrt{3}\), and \(\cot\frac{29\pi}{6}=-\sqrt{3}\).

Answer:

A. \(\cot\frac{29\pi}{6}=-\sqrt{3}\)