Question

determine which of the following is equivalent to the given expression.
\\(\cos\left(\frac{7\pi}{6}\
ight)\\)

Explanation:

Step1: Rewrite the angle

We know that $\frac{7\pi}{6}=\pi+\frac{\pi}{6}$. So, $\cos(\frac{7\pi}{6})=\cos(\pi + \frac{\pi}{6})$.

Step2: Apply the cosine - sum formula

The formula for $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A = \pi$ and $B=\frac{\pi}{6}$. We know that $\cos(\pi)=- 1$, $\sin(\pi)=0$, $\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}$ and $\sin(\frac{\pi}{6})=\frac{1}{2}$. Then $\cos(\pi+\frac{\pi}{6})=\cos(\pi)\cos(\frac{\pi}{6})-\sin(\pi)\sin(\frac{\pi}{6})=(-1)\times\frac{\sqrt{3}}{2}-0\times\frac{1}{2}=-\frac{\sqrt{3}}{2}$.

Answer:

$-\frac{\sqrt{3}}{2}$