Question

use the unit circle to find the value of tan $\frac{3pi}{4}$ and periodic properties of trigonometric functions to find the value of tan $\frac{15pi}{4}$. select the correct choice below and fill in any answer boxes in your choice. a. tan $\frac{3pi}{4}=$ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined.

Explanation:

Step1: Recall tangent formula

The formula for $\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta = \frac{3\pi}{4}$, on the unit - circle, the coordinates of the point corresponding to the angle $\frac{3\pi}{4}$ are $(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. Since $\sin\theta$ is the $y$ - coordinate and $\cos\theta$ is the $x$ - coordinate, $\sin\frac{3\pi}{4}=\frac{\sqrt{2}}{2}$ and $\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}$.

Step2: Calculate $\tan\frac{3\pi}{4}$

$\tan\frac{3\pi}{4}=\frac{\sin\frac{3\pi}{4}}{\cos\frac{3\pi}{4}}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=- 1$.

Step3: Use the periodic property of tangent

The period of the tangent function $y = \tan x$ is $\pi$. We can write $\frac{15\pi}{4}=4\pi-\frac{\pi}{4}$. Since $\tan(x + k\pi)=\tan x$ for any real number $x$ and integer $k$, $\tan\frac{15\pi}{4}=\tan(4\pi-\frac{\pi}{4})$. And $\tan(4\pi-\frac{\pi}{4})=\tan(-\frac{\pi}{4})$. Since $\tan(-x)=-\tan x$, $\tan(-\frac{\pi}{4})=-\tan\frac{\pi}{4}=-1$.

Answer:

A. $\tan\frac{3\pi}{4}=-1$