3. if the angle ( \theta=\frac{7 pi}{4} ) is in standard position on the unit circle, which ordered pair…

Explanation:

Step1: Determine the quadrant of the angle

The angle (\theta=\frac{7\pi}{4}). Since (2\pi-\frac{\pi}{4}=\frac{7\pi}{4}), the angle (\frac{7\pi}{4}) is in the fourth - quadrant. In the fourth - quadrant, the (x) - coordinate is positive and the (y) - coordinate is negative.

Step2: Use the unit - circle definitions

For a unit - circle (x = \cos\theta) and (y=\sin\theta). We know that (\cos\frac{7\pi}{4}=\cos(2\pi - \frac{\pi}{4})) and (\sin\frac{7\pi}{4}=\sin(2\pi-\frac{\pi}{4})). Using the trigonometric identities (\cos(A - B)=\cos A\cos B+\sin A\sin B) and (\sin(A - B)=\sin A\cos B-\cos A\sin B) with (A = 2\pi) and (B=\frac{\pi}{4}), we have (\cos(2\pi-\frac{\pi}{4})=\cos\frac{\pi}{4}) and (\sin(2\pi - \frac{\pi}{4})=-\sin\frac{\pi}{4}). Since (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}) and (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), then (x=\frac{\sqrt{2}}{2}) and (y =-\frac{\sqrt{2}}{2}).

Answer:

(\left(\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right))