find the exact value of $cos\frac{3pi}{4}$. $cos\frac{3pi}{4}=$

Explanation:

Step1: Rewrite the angle

We know that $\frac{3\pi}{4}=\pi - \frac{\pi}{4}$. So, $\cos\frac{3\pi}{4}=\cos(\pi-\frac{\pi}{4})$.

Step2: Apply the cosine - difference formula

The formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$. Here $A = \pi$ and $B=\frac{\pi}{4}$. Also, $\cos(A - B)=-\cos B$ when $A=\pi$. So, $\cos(\pi-\frac{\pi}{4})=-\cos\frac{\pi}{4}$.

Step3: Recall the value of $\cos\frac{\pi}{4}$

We know that $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. So, $-\cos\frac{\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

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