find the exact value of cos 5π/4. cos 5π/4 =

Explanation:

Step1: Rewrite the angle

We know that $\frac{5\pi}{4}=\pi+\frac{\pi}{4}$.

Step2: Use the cosine - addition formula

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

Step3: Substitute known values

We know that $\cos\pi=- 1$, $\sin\pi = 0$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$. Then $\cos(\pi+\frac{\pi}{4})=(-1)\times\frac{\sqrt{2}}{2}-0\times\frac{\sqrt{2}}{2}=-\frac{\sqrt{2}}{2}$.

Answer:

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