Question

If \(tan(A-B)=\frac {1}{\sqrt 3},\ cos \ A=\frac 12 \) then \(∠B=\) ____ .

• A. \(\frac {2\pi}{3}\)
• B. \(\frac {\pi}{4}\)
• C. \(\frac {\pi}{6}\)
• D. \(\frac {\pi}{3}\)

Answer 1

The Correct Option is C

To solve the problem, we are given:

• \( \tan(A - B) = \frac{1}{\sqrt{3}} \)
• \( \cos A = \frac{1}{2} \)

We are to find \( \angle B \).

1. Determine Angle A:
From \( \cos A = \frac{1}{2} \), we know:
\[ A = \cos^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{3} \]

2. Use the Value of \( \tan(A - B) \):
Given \( \tan(A - B) = \frac{1}{\sqrt{3}} \), we recognize that:
\[ \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \Rightarrow A - B = \frac{\pi}{6} \]

3. Solve for \( B \):
\[ A - B = \frac{\pi}{6} \Rightarrow \frac{\pi}{3} - B = \frac{\pi}{6} \]
\[ B = \frac{\pi}{3} - \frac{\pi}{6} = \frac{\pi}{6} \]

Final Answer:
The value of \( \angle B \) is \( \frac{\pi}{6} \).