Question

Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) + \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) + \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \)

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

Hint:

When adding inverse trigonometric functions, ensure that you use the correct reference angles and quadrant considerations.

Answer 1

The Correct Option is B

Step 1: Evaluate \( \sin^{-1}\left(\frac{1}{2}\right) \).
We know that \( \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \).
Step 2: Evaluate \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) \).
We know that \( \cos^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{6} \).
Step 3: Evaluate \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \).
We know that \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) = \frac{5\pi}{6} \).
Step 4: Add the results.
\[ \frac{\pi}{6} + \frac{\pi}{6} + \frac{5\pi}{6} = \pi \] Step 5: Conclusion.
Hence, the value is \( \pi \).