Question

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

quadrant i
quadrant ii
quadrant iv
along an axis

Explanation:

Step1: Recall sine - angle relationship

The sine function \(y = \sin\theta\) has the property \(\sin\theta=\sin(\pi - \theta)\) and \(\sin\theta=\sin(2k\pi+\theta)\) for \(k\in\mathbb{Z}\). The angle \(\theta=\frac{5\pi}{4}\) is in Quadrant III (\(\pi<\frac{5\pi}{4}<\frac{3\pi}{2}\)).

Step2: Find equivalent - sine angles

We know that \(\sin\theta=\sin(\pi - \theta)\). For \(\theta = \frac{5\pi}{4}\), \(\pi-\frac{5\pi}{4}=-\frac{\pi}{4}\), and also \(\sin\theta=\sin(2\pi-\theta)\). \(\sin\frac{5\pi}{4}=\sin(\frac{5\pi}{4}- 2\pi)=\sin(-\frac{3\pi}{4})\). Another way is to use the identity \(\sin\theta=\sin(\pi - \theta)\). If \(\theta=\frac{5\pi}{4}\), then \(\sin\frac{5\pi}{4}=\sin(\pi-\frac{5\pi}{4})=\sin(-\frac{\pi}{4})\). The angle with the same sine value as \(\frac{5\pi}{4}\) is \(\pi-\frac{5\pi}{4}=-\frac{\pi}{4}\) (or \(\frac{7\pi}{4}\) which is in Quadrant IV since \(\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi\)).

Answer:

C. Quadrant IV