Question

evaluate. write your answer in simplified, rationalized form. do not round.
sec(π/4) =

Explanation:

Step1: Use the reciprocal identity

We know that \(\sec\theta=\frac{1}{\cos\theta}\). So, \(\sec(\frac{\pi}{4})=\frac{1}{\cos(\frac{\pi}{4})}\).

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

The value of \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\).

Step3: Substitute and simplify

Substitute \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\) into \(\sec(\frac{\pi}{4})=\frac{1}{\cos(\frac{\pi}{4})}\), we get \(\sec(\frac{\pi}{4})=\frac{1}{\frac{\sqrt{2}}{2}}\). Using the rule \(\frac{a}{\frac{b}{c}}=\frac{ac}{b}\), then \(\frac{1}{\frac{\sqrt{2}}{2}}=\frac{2}{\sqrt{2}}\). Rationalize the denominator: \(\frac{2}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\sqrt{2}\).

Answer:

\(\sqrt{2}\)