Question

find the exact value of the following expression without using a calculator. sec 3π/4

Explanation:

Step1: Recall the secant formula

\(\sec\theta=\frac{1}{\cos\theta}\), so \(\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}\).

Step2: Find the value of \(\cos\frac{3\pi}{4}\)

\(\frac{3\pi}{4}\) is in the second quadrant. \(\cos\frac{3\pi}{4}=-\cos(\pi - \frac{3\pi}{4})=-\cos\frac{\pi}{4}\). Since \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), then \(\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}\).

Step3: Calculate \(\sec\frac{3\pi}{4}\)

Substitute \(\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}\) into \(\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}\), we get \(\sec\frac{3\pi}{4}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}\).

Answer:

\(-\sqrt{2}\)