Question

use the unit circle to find the value of cos(π/4) and periodic properties of trigonometric functions to find the value of cos(17π/4). select the correct choice below and fill in any answer boxes in your choice. a. cos(π/4)=□ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined.

Explanation:

Step1: Recall unit - circle values for $\cos(\frac{\pi}{4})$

On the unit circle, for the angle $\theta=\frac{\pi}{4}$, the coordinates of the corresponding point are $(\cos\theta,\sin\theta)=(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2})$. So, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Step2: Use the periodicity of the cosine function

The cosine function has a period of $2\pi$, i.e., $\cos(x + 2k\pi)=\cos(x)$ for any real - number $x$ and integer $k$. We can rewrite $\frac{17\pi}{4}$ as $\frac{17\pi}{4}=4\pi+\frac{\pi}{4}$. Since $4\pi = 2\times2\pi$, then $\cos\frac{17\pi}{4}=\cos(4\pi+\frac{\pi}{4})$. By the periodicity property, $\cos(4\pi+\frac{\pi}{4})=\cos\frac{\pi}{4}$.

Answer:

A. $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$