Question
Find the exact value of the expression. \( \sec \left(\frac{\pi}{4}\right)=\square \)
Ask by Bartlett Wang. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The exact value of \( \sec \left(\frac{\pi}{4}\right) \) is \( \sqrt{2} \).
Solution
Calculate or simplify the expression \( \sec(\frac{\pi}{4}) \).
Calculate the value by following steps:
- step0: Calculate:
\(\sec\left(\frac{\pi }{4}\right)\)
- step1: Calculate the trigonometric value:
\(\sqrt{2}\)
The exact value of the expression \( \sec \left(\frac{\pi}{4}\right) \) is \( \sqrt{2} \).
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Beyond the Answer
The value of \( \sec \left(\frac{\pi}{4}\right) \) can be found by recalling that secant is the reciprocal of cosine. Since \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we have: \[ \sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}. \] Thus, \( \sec \left(\frac{\pi}{4}\right) = \sqrt{2} \).