Question

use reference angles to find the exact value of the following expression. tan(13π/3) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. tan(13π/3)=□ (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize the denominator.) b. the answer is undefined.

Explanation:

Step1: Find an equivalent angle.

First, find an equivalent angle for $\frac{13\pi}{3}$ within one - full rotation ($2\pi$). We know that $\frac{13\pi}{3}=4\pi+\frac{\pi}{3}$. Since $4\pi$ is a multiple of $2\pi$, $\tan(\frac{13\pi}{3})=\tan(4\pi + \frac{\pi}{3})$. According to the periodicity of the tangent function $\tan(x + n\cdot2\pi)=\tan(x)$ where $n\in\mathbb{Z}$, so $\tan(\frac{13\pi}{3})=\tan(\frac{\pi}{3})$.

Step2: Recall the value of tangent for the reference angle.

The reference angle $\frac{\pi}{3}$ is in the first - quadrant. We know that for an angle $\theta=\frac{\pi}{3}$ in a right - triangle, if the opposite side to the angle $\theta$ is $\sqrt{3}$ and the adjacent side is $1$, then $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. So, $\tan(\frac{\pi}{3})=\sqrt{3}$.

Answer:

A. $\tan\frac{13\pi}{3}=\sqrt{3}$