find the exact values below. if applicable, click on \undefined\. tan 7π/6 = csc 7π/6 =

Answer:

1. $\tan\frac{7\pi}{6}=\frac{\sqrt{3}}{3}$
2. $\csc\frac{7\pi}{6}=- 2$

Explanation:

Step1: Rewrite the angle

Rewrite $\frac{7\pi}{6}$ as $\pi+\frac{\pi}{6}$.

Step2: Use trigonometric identities for $\tan(A + B)$

For $\tan(A + B)=\frac{\tan A+\tan B}{1 - \tan A\tan B}$, when $A=\pi$ and $B = \frac{\pi}{6}$, $\tan(\pi+\frac{\pi}{6})=\tan\frac{\pi}{6}$ since $\tan\pi = 0$. And $\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}$.

Step3: Recall the definition of cosecant

Recall that $\csc\theta=\frac{1}{\sin\theta}$. First, find $\sin\frac{7\pi}{6}=\sin(\pi+\frac{\pi}{6})$. Using the identity $\sin(A + B)=\sin A\cos B+\cos A\sin B$ with $A=\pi$ and $B=\frac{\pi}{6}$, we have $\sin(\pi+\frac{\pi}{6})=-\sin\frac{\pi}{6}=-\frac{1}{2}$. Then $\csc\frac{7\pi}{6}=\frac{1}{\sin\frac{7\pi}{6}}=-2$.