an angle that shares the same sine value of an angle that measures $\frac{5pi}{4}$ radians is located…

Explanation:

Step1: Recall sine - angle relationship

The sine function has the property $\sin(\pi - \theta)=\sin\theta$ and $\sin(2k\pi+\theta)=\sin\theta,k\in\mathbb{Z}$. Given $\theta = \frac{5\pi}{4}$, which is in Quadrant III ($\pi<\frac{5\pi}{4}<\frac{3\pi}{2}$). We know that $\sin(\frac{5\pi}{4})=\sin(\pi + \frac{\pi}{4})=-\sin(\frac{\pi}{4})$. Also, $\sin(\pi-\frac{5\pi}{4})=\sin(-\frac{\pi}{4})=-\sin(\frac{\pi}{4})$. The angle $\pi - \frac{5\pi}{4}=-\frac{\pi}{4}$ which is equivalent to $2\pi-\frac{\pi}{4}=\frac{7\pi}{4}$ (in the range $[0, 2\pi)$) and is in Quadrant IV. Another way is to use the identity $\sin\alpha=\sin\beta$ implies $\alpha=\beta + 2k\pi$ or $\alpha=\pi-\beta + 2k\pi,k\in\mathbb{Z}$. For $\beta=\frac{5\pi}{4}$, $\pi-\frac{5\pi}{4}=-\frac{\pi}{4}$ and its coterminal angle in $[0,2\pi)$ is $2\pi-\frac{\pi}{4}=\frac{7\pi}{4}$ which lies in Quadrant IV.

Answer:

Quadrant IV