Question

find the exact values of the six trigonometric functions of the given angle. do not use a calculator.
-\frac{7\pi}{6}
select the correct choice below and fill in any answer boxes within your choice.
a. \\(\sin(-\frac{7\pi}{6}) = \square\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
b. the function value is undefined.

Explanation:

Step1: Rewrite the angle

We know that $-\frac{7\pi}{6}=\pi-\frac{\pi}{6}$.

Step2: Use the sine - angle formula

$\sin(-\frac{7\pi}{6})=\sin(\pi - \frac{\pi}{6})$. According to the formula $\sin(A - B)=\sin A\cos B-\cos A\sin B$, when $A = \pi$ and $B=\frac{\pi}{6}$, we have $\sin(\pi-\alpha)=\sin\alpha$. So $\sin(-\frac{7\pi}{6})=\sin\frac{\pi}{6}$.

Step3: Find the value of $\sin\frac{\pi}{6}$

We know that $\sin\frac{\pi}{6}=\frac{1}{2}$.

Answer:

A. $\sin(-\frac{7\pi}{6})=\frac{1}{2}$