without using a calculator, compute the sine, cosine, and tangent of 7π/4 by using the reference angle…

Explanation:

Step1: Determine the quadrant

Since (2\pi = \frac{8\pi}{4}) and (\frac{3\pi}{2}=\frac{6\pi}{4}), and (\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi), the angle (\frac{7\pi}{4}) is in the fourth - quadrant. So the answer for the quadrant is 4.

Step2: Find the reference angle

The reference angle (\theta_{r}) for an angle (\theta) in the fourth - quadrant is given by (\theta_{r}=2\pi-\theta). Here (\theta = \frac{7\pi}{4}), so (\theta_{r}=2\pi-\frac{7\pi}{4}=\frac{8\pi - 7\pi}{4}=\frac{\pi}{4}).

Step3: Find the sine value

In the fourth - quadrant, (\sin\theta<0). We know that (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}), so (\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}).

Step4: Find the cosine value

In the fourth - quadrant, (\cos\theta>0). Since (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}), then (\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}).

Step5: Find the tangent value

(\tan\theta=\frac{\sin\theta}{\cos\theta}), so (\tan\frac{7\pi}{4}=\frac{\sin\frac{7\pi}{4}}{\cos\frac{7\pi}{4}}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=- 1).

Answer:

What is the reference angle? (\frac{\pi}{4}) radians. What quadrant is this angle in? 4 (\sin(\frac{7\pi}{4})=-\frac{\sqrt{2}}{2}) (\cos(\frac{7\pi}{4})=\frac{\sqrt{2}}{2}) (\tan(\frac{7\pi}{4})=-1)