Question

(a) Find the value of \( \sin^{-1}\left( -\frac{1}{2} \right) + \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) + \cot^{-1}\left( \tan \frac{4\pi}{3} \right) \).

Hint:

For inverse trigonometric functions: - \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \) have specific ranges, ensuring their outputs correspond to the correct quadrant. - Use trigonometric identities to simplify the inverse tangent expressions.

Answer 1

Step 1: {Evaluate \( \sin^{-1}\left( -\frac{1}{2} \right) \)}
The angle whose sine is \( -\frac{1}{2} \) is \( -\frac{\pi}{6} \), hence: \[ \sin^{-1}\left( -\frac{1}{2} \right) = -\frac{\pi}{6}. \] Step 2: {Evaluate \( \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) \)}
The angle whose cosine is \( -\frac{\sqrt{3}}{2} \) is \( \frac{5\pi}{6} \), so: \[ \cos^{-1}\left( -\frac{\sqrt{3}}{2} \right) = \frac{5\pi}{6}. \]