Question

write the expression as a function of x, with no angle measure involved.

cos (\frac{5\pi}{6}+x)

cos (\frac{5\pi}{6}+x)=\square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the cosine addition formula

The cosine addition formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A=\frac{5\pi}{6}$ and $B = x$.
So, $\cos(\frac{5\pi}{6}+x)=\cos\frac{5\pi}{6}\cos x-\sin\frac{5\pi}{6}\sin x$.

Step2: Find the values of $\cos\frac{5\pi}{6}$ and $\sin\frac{5\pi}{6}$

We know that $\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$ and $\sin\frac{5\pi}{6}=\frac{1}{2}$.
Substitute these values into the expression:
$\cos(\frac{5\pi}{6}+x)=-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x$.

Answer:

$-\frac{\sqrt{3}}{2}\cos x-\frac{1}{2}\sin x$