Question

given the following unit circle, rotate the green dot to the appropriate angle and then find the exact value of the function.
\\(\sin \frac{7\pi}{4}\\)

Explanation:

Step1: Determine the quadrant of \( \frac{7\pi}{4} \)

The angle \( \frac{7\pi}{4} \) is in the fourth quadrant since \( \frac{3\pi}{2} < \frac{7\pi}{4} < 2\pi \).

Step2: Find the reference angle

The reference angle \( \theta' \) for an angle \( \theta \) in the fourth quadrant is \( 2\pi - \theta \). So, \( \theta' = 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4} \).

Step3: Determine the sign of \( \sin \frac{7\pi}{4} \)

In the fourth quadrant, the sine function is negative (since sine corresponds to the y - coordinate on the unit circle, and in the fourth quadrant, y - coordinates are negative).

Step4: Recall the value of \( \sin \frac{\pi}{4} \)

We know that \( \sin \frac{\pi}{4}=\frac{\sqrt{2}}{2} \).

Step5: Find \( \sin \frac{7\pi}{4} \)

Using the reference angle and the sign, \( \sin \frac{7\pi}{4}=-\sin \frac{\pi}{4}=-\frac{\sqrt{2}}{2} \).

Answer:

\( -\frac{\sqrt{2}}{2} \)