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 angle on unit - circle

The angle $\frac{7\pi}{4}$ is equivalent to $2\pi-\frac{\pi}{4}$. On the unit - circle, the coordinates of the point corresponding to an angle $\theta$ are $(\cos\theta,\sin\theta)$. The angle $\frac{7\pi}{4}$ is in the fourth quadrant, and its reference angle is $\frac{\pi}{4}$.

Step2: Recall sine value for reference angle

For an angle $\theta$ with reference angle $\alpha$, $\sin\theta$ has a value related to the $y$ - coordinate of the point on the unit - circle. For $\theta=\frac{7\pi}{4}$ and reference angle $\alpha = \frac{\pi}{4}$, and since in the fourth quadrant the $y$ - coordinate is negative, and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, we have $\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}$.

Step3: Analyze $\sin\frac{15\pi}{4}$

We know that $\frac{15\pi}{4}=4\pi-\frac{\pi}{4}$. Since the sine function has a period of $2\pi$, $\sin(4\pi - \frac{\pi}{4})=\sin(-\frac{\pi}{4})$. And $\sin(-\theta)=-\sin\theta$, so $\sin\frac{15\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

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