Question

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)