Question

If \(\tan\left(\alpha - \frac{\pi}{12}\right) = \frac{1}{\sqrt{3}}\), where \(0 < \alpha < \frac{\pi}{2}\), then the value of \(\alpha\) is equal to

• A. \(\frac{\pi}{3}\)
• B. \(\frac{4\pi}{9}\)
• C. \(\frac{\pi}{4}\)
• D. \(\frac{\pi}{6}\)
• E. \(\frac{\pi}{8}\)

Hint:

\(\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
\(\frac{1}{\sqrt{3}} = \tan \frac{\pi}{6}\). So \(\alpha - \frac{\pi}{12} = \frac{\pi}{6} + n\pi\). Since \(0<\alpha<\frac{\pi}{2}\), take \(n = 0\).

Step 2: Detailed Explanation:
\(\alpha - \frac{\pi}{12} = \frac{\pi}{6} \Rightarrow \alpha = \frac{\pi}{6} + \frac{\pi}{12} = \frac{2\pi}{12} + \frac{\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4}\)

Step 3: Final Answer:
\(\alpha = \frac{\pi}{4}\).