evaluate without using a calculator by using ratios in a reference triangle. sec 3π/4

Explanation:

Step1: Recall the definition of secant

(\sec\theta=\frac{1}{\cos\theta})

Step2: Find the reference angle and quadrant for (\frac{3\pi}{4})

The angle (\frac{3\pi}{4}) is in the second quadrant. The reference angle (\alpha=\pi - \frac{3\pi}{4}=\frac{\pi}{4})

Step3: Determine the value of (\cos\frac{3\pi}{4})

In the second quadrant, (\cos\theta<0). Since (\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}), then (\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2})

Step4: Calculate (\sec\frac{3\pi}{4})

Using (\sec\theta=\frac{1}{\cos\theta}), we have (\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2})

Answer:

(-\sqrt{2}) (corresponding to the second option in the multiple - choice list)