find the exact value of $\\cos\\frac{5\\pi}{6}$, $\\cos\\frac{5\\pi}{6}=\\square$

Explanation:

Step1: Use the formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$

We know that $\frac{5\pi}{6}=\pi-\frac{\pi}{6}$. So, $\cos\frac{5\pi}{6}=\cos(\pi - \frac{\pi}{6})$. Using the formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$ with $A = \pi$ and $B=\frac{\pi}{6}$, we get $\cos(\pi-\frac{\pi}{6})=\cos\pi\cos\frac{\pi}{6}+\sin\pi\sin\frac{\pi}{6}$. Since $\cos\pi=- 1$, $\sin\pi = 0$, and $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$, $\sin\frac{\pi}{6}=\frac{1}{2}$.

Step2: Substitute the values

Substitute the values into the formula: [ \begin{align*} \cos(\pi-\frac{\pi}{6})&=(-1)\times\frac{\sqrt{3}}{2}+0\times\frac{1}{2}\ &=-\frac{\sqrt{3}}{2} \end{align*} ]

Answer:

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