Question

use the unit circle to find the value of cos(7π/4) and periodic properties of trigonometric functions to find the value of cos(15π/4). select the correct choice below and fill in any answer boxes in your choice. a. cos(7π/4)=□ (type an exact answer, using radicals as needed. simplify your answer. rationalize the denominator.) b. the solution is undefined.

Explanation:

Step1: Locate angle on unit - circle

The angle $\frac{7\pi}{4}$ is equivalent to $2\pi-\frac{\pi}{4}$. On the unit - circle, the angle $\frac{7\pi}{4}$ has a terminal side in the fourth quadrant. The coordinates of the point on the unit - circle corresponding to the angle $\theta$ are $(\cos\theta,\sin\theta)$. For $\theta = \frac{7\pi}{4}$, the coordinates are $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$. So, $\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}$.

Step2: Use periodicity for $\frac{15\pi}{4}$

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 write $\frac{15\pi}{4}=4\pi-\frac{\pi}{4}$. Since $4\pi = 2\times2\pi$, $\cos\frac{15\pi}{4}=\cos(4\pi-\frac{\pi}{4})=\cos(-\frac{\pi}{4})$. And since $\cos(-x)=\cos(x)$ for any real number $x$, $\cos\frac{15\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Answer:

A. $\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}$