evaluate without a calculator.\ncos $\frac{7pi}{4}$ = ? $\frac{sqrt{}}{}$\nenter + or -

Explanation:

Step1: Rewrite the angle

We know that $\frac{7\pi}{4}=2\pi-\frac{\pi}{4}$.

Step2: Use the cosine - angle formula

Since $\cos(A - B)=\cos A\cos B+\sin A\sin B$, and $\cos(2\pi-\alpha)=\cos2\pi\cos\alpha+\sin2\pi\sin\alpha$. Also, $\cos2\pi = 1$ and $\sin2\pi=0$, so $\cos(2\pi - \alpha)=\cos\alpha$. Then $\cos\frac{7\pi}{4}=\cos(2\pi-\frac{\pi}{4})=\cos\frac{\pi}{4}$.

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

We know that for an angle $\theta=\frac{\pi}{4}$ in a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Answer:

$+\frac{\sqrt{2}}{2}$