Question

complete the parts below.
(a) two unit - circles are shown. sketch the requested angles in standard position.
sketch the angle -\\(\frac{7\pi}{4}\\) radians. sketch the angle -\\(\frac{7\pi}{4}-4\pi\\) radians.
(b) find the following. use exact values and not decimal approximations.
\\(\sin(-\frac{7\pi}{4}) =\\)
\\(\sin(-\frac{7\pi}{4}-4\pi)=\\)
(c) choose the correct statement.
for any angle \\(\theta\\) measured in radians, it is always the case that \\(\sin(\theta)=\sin(\theta - 4\pi)\\).
for some but not all angles \\(\theta\\) measured in radians, we have \\(\sin(\theta)=\sin(\theta - 4\pi)\\).
for any angle \\(\theta\\) measured in radians, it is always the case that \\(\sin(\theta)=\sin(\theta-4\pi)\\).

Explanation:

Step1: Recall sine - angle properties

The sine function has a period of \(2\pi\), i.e., \(\sin(x)=\sin(x + 2k\pi)\) for any real - number \(x\) and integer \(k\).

Step2: Find \(\sin(-\frac{7\pi}{4})\)

We know that \(\sin(-\frac{7\pi}{4})=\sin(-2\pi+\frac{\pi}{4})\). Since the period of the sine function is \(2\pi\), \(\sin(-2\pi+\frac{\pi}{4})=\sin(\frac{\pi}{4})\). And \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\).

Step3: Find \(\sin(-\frac{7\pi}{4}-4\pi)\)

\(\sin(-\frac{7\pi}{4}-4\pi)=\sin(-\frac{7\pi}{4})\) because \(4\pi = 2\times2\pi\) and the period of the sine function is \(2\pi\). So \(\sin(-\frac{7\pi}{4}-4\pi)=\frac{\sqrt{2}}{2}\).

Step4: Analyze the statement about the sine function

Since the period of \(y = \sin\theta\) is \(2\pi\), for any angle \(\theta\) measured in radians, \(\sin\theta=\sin(\theta + 2k\pi)\) where \(k\in\mathbb{Z}\). When \(k = - 2\), \(\sin\theta=\sin(\theta-4\pi)\).

Answer:

\(\sin(-\frac{7\pi}{4})=\frac{\sqrt{2}}{2}\)
\(\sin(-\frac{7\pi}{4}-4\pi)=\frac{\sqrt{2}}{2}\)
The correct statement is: For any angle \(\theta\) measured in radians, it is always the case that \(\sin\theta=\sin(\theta - 4\pi)\)