Question

The value of \( \sin^{-1} \left( -\frac{1}{2} \right) + \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \) is

• A. \( \cos^{-1} \left( \frac{1}{2} \right) \)
• B. \( \sin^{-1} \left( -\frac{1}{2} \right) \)
• C. \( \cos^{-1} \left( -\frac{1}{2} \right) \)
• D. \( \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \)

Hint:

When adding inverse trigonometric functions, make sure to simplify the angles and check the values using known trigonometric identities.

Answer 1

The Correct Option is C

Step 1: Understanding the trigonometric inverses.
We are given the sum of two inverse trigonometric functions: \( \sin^{-1} \left( -\frac{1}{2} \right) \) and \( \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \). The value of each of these functions corresponds to specific angles.

Step 2: Calculating the inverse functions.
- \( \sin^{-1} \left( -\frac{1}{2} \right) = -\frac{\pi}{6} \) (since \( \sin \left( -\frac{\pi}{6} \right) = -\frac{1}{2} \)).
- \( \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) = \frac{5\pi}{6} \) (since \( \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2} \)).
Step 3: Summing the results.
Now, we calculate the sum: \[ -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \] This is equivalent to \( \cos^{-1} \left( -\frac{1}{2} \right) \), which gives us option (C).

Step 4: Conclusion.
Thus, the correct answer is \( \cos^{-1} \left( -\frac{1}{2} \right) \), which makes option (C) the correct answer.