Question
\sin \left(\frac{7\pi }{4}\right)
Answers
GPT-4o mini
To find \(\sin \left(\frac{7\pi}{4}\right)\), we can recognize that \(\frac{7\pi}{4}\) is in the fourth quadrant of the unit circle.
The angle \(\frac{7\pi}{4}\) can also be expressed as:
\[
\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}
\]
This means it is \(45^\circ\) (or \(\frac{\pi}{4}\)) below the x-axis.
In the fourth quadrant, the sine function is negative. The reference angle is \(\frac{\pi}{4}\), where:
\[
\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Thus:
\[
\sin \left(\frac{7\pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Therefore, the value of \(\sin \left(\frac{7\pi}{4}\right)\) is:
\[
\boxed{-\frac{\sqrt{2}}{2}}
\]
The angle \(\frac{7\pi}{4}\) can also be expressed as:
\[
\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}
\]
This means it is \(45^\circ\) (or \(\frac{\pi}{4}\)) below the x-axis.
In the fourth quadrant, the sine function is negative. The reference angle is \(\frac{\pi}{4}\), where:
\[
\sin \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Thus:
\[
\sin \left(\frac{7\pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]
Therefore, the value of \(\sin \left(\frac{7\pi}{4}\right)\) is:
\[
\boxed{-\frac{\sqrt{2}}{2}}
\]