Given:
[tex]\frac{7\pi}{4}[/tex]
To Determine: The value of the given on a unit circle
Solution
For sin 7pi/4, the angle 7pi/4 lies between 3pi/2 and 2pi (Fourth Quadrant)
[tex]\begin{gathered} sin(\frac{7\pi}{4}) \\ Note:\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi \\ Therefore,the\text{ angle lies in the 4th quadrant} \end{gathered}[/tex]
This is as shown below
In a simplified form, we have it as shown in the image below
[tex]The\text{ coordinate of sin}\frac{7\pi}{4}=(0.7071,-0.7071),or(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}})[/tex]
Note that sine is negative in the fourth quadrant
[tex]\begin{gathered} sin(\frac{7\pi}{4})=-sin(2\pi-\frac{7\pi}{4}) \\ sin(\frac{7\pi}{4})=-sin(\frac{8\pi-7\pi}{4}) \\ sin(\frac{7\pi}{4})=-sin(\frac{\pi}{4}) \\ sin(\frac{7\pi}{4})=-\frac{1}{\sqrt{2}},or,-\frac{\sqrt{2}}{2} \end{gathered}[/tex]
Hence,
[tex]\begin{gathered} sin(\frac{7\pi}{4})=-\frac{1}{\sqrt{2}},or \\ sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2} \end{gathered}[/tex]
