Question

use the unit circle to find the value of tan(7π/4) and periodic properties of trigonometric functions to find the value of tan(27π/4). select the correct choice below and fill in any answer boxes in your choice. a. tan(7π/4)=□ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined. select the correct choice below and fill in any answer boxes in your choice. a. tan(27π/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

$\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta = \frac{7\pi}{4}$, on the unit - circle, $\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}$ and $\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}$.

Step2: Calculate $\tan\frac{7\pi}{4}$

$\tan\frac{7\pi}{4}=\frac{\sin\frac{7\pi}{4}}{\cos\frac{7\pi}{4}}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=- 1$.

Step3: Use periodic property of tangent

The period of the tangent function is $\pi$, i.e., $\tan(x + n\pi)=\tan(x)$ for any real number $x$ and integer $n$. We can write $\frac{27\pi}{4}=6\pi+\frac{3\pi}{4}$. So, $\tan\frac{27\pi}{4}=\tan(6\pi+\frac{3\pi}{4})=\tan\frac{3\pi}{4}$.

Step4: Calculate $\tan\frac{3\pi}{4}$

On the unit - circle, for $\theta=\frac{3\pi}{4}$, $\sin\frac{3\pi}{4}=\frac{\sqrt{2}}{2}$ and $\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}$. Then $\tan\frac{3\pi}{4}=\frac{\sin\frac{3\pi}{4}}{\cos\frac{3\pi}{4}}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=-1$.

Answer:

A. $\tan\frac{7\pi}{4}=-1$
A. $\tan\frac{27\pi}{4}=-1$