Evaluate: sin-1(−√3/2) + cosec-1(−2/√3)
Solution:
Step 1: Evaluate sin⁻¹(−√3/2)
\[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]
(Principal range: \([- \pi/2, \pi/2]\))
Step 2: Evaluate cosec⁻¹(−2/√3)
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) \Rightarrow \sin y = -\frac{\sqrt{3}}{2} \]
\[ \Rightarrow y = -\frac{\pi}{3} \]
(Principal range: \([-\pi/2,0) \cup (0,\pi/2]\))
Step 3: Add values
\[ -\frac{\pi}{3} + \left(-\frac{\pi}{3}\right) = -\frac{2\pi}{3} \]
Final Answer:
Value = \[ -\frac{2\pi}{3} \]