Question

use reference angles to find the exact value of the following expression. do not use a calculator. cot(23π/3)
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. cot(23π/3) = (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: Rewrite the angle

First, rewrite $\frac{23\pi}{3}$ as $\frac{24\pi - \pi}{3}=8\pi-\frac{\pi}{3}$. Since $8\pi$ is a multiple of $2\pi$, $\cot(\frac{23\pi}{3})=\cot(8\pi - \frac{\pi}{3})$.

Step2: Use cotangent - periodicity and identity

The cotangent function has a period of $\pi$, and $\cot(A - B)=\frac{\cot A\cot B + 1}{\cot B-\cot A}$. For $\cot(8\pi - \frac{\pi}{3})$, because $\cot(2k\pi + x)=\cot x$ and $\cot(-x)=-\cot x$ ($k\in\mathbb{Z}$), we have $\cot(8\pi - \frac{\pi}{3})=\cot(-\frac{\pi}{3})=-\cot\frac{\pi}{3}$.

Step3: Find the value of $\cot\frac{\pi}{3}$

We know that $\cot\theta=\frac{\cos\theta}{\sin\theta}$, and for $\theta = \frac{\pi}{3}$, $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$. So $\cot\frac{\pi}{3}=\frac{\cos\frac{\pi}{3}}{\sin\frac{\pi}{3}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}$.

Step4: Calculate the final value

Then $-\cot\frac{\pi}{3}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$.

Answer:

A. $\cot\frac{23\pi}{3}=-\frac{\sqrt{3}}{3}$