Question

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

Explanation:

Step1: Analyze the angle $\frac{7\pi}{4}$ on unit - circle

The angle $\frac{7\pi}{4}=2\pi-\frac{\pi}{4}$. In the unit - circle, for an angle $\theta$ in standard position, $\sin\theta$ is the $y$ - coordinate of the point where the terminal side of the angle intersects the unit circle. The angle $\frac{7\pi}{4}$ is in the fourth quadrant and is coterminal with $-\frac{\pi}{4}$. The coordinates of the point on the unit circle corresponding to $\frac{7\pi}{4}$ are $(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})$. So, $\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}$.

Step2: Analyze the angle $\frac{15\pi}{4}$ using periodicity

The period of the sine function is $2\pi$, i.e., $\sin(x + 2k\pi)=\sin x$ for any real number $x$ and integer $k$. We can write $\frac{15\pi}{4}=4\pi-\frac{\pi}{4}$. Since $\sin(4\pi-\frac{\pi}{4})=\sin(-\frac{\pi}{4})$ (because of the periodicity of the sine function with period $2\pi$), and $\sin(-\alpha)=-\sin\alpha$, then $\sin\frac{15\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

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