Find the Value of sin⁻¹(-√3/2) + cos⁻¹(-1/2)
Given Expression
\[
\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) + \cos^{-1}\left(-\frac{1}{2}\right)
\]
Step 1: Use Principal Values
The principal value range of:
\( \sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}] \)
\( \cos^{-1}x \in [0, \pi] \)
\( \sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}] \)
\( \cos^{-1}x \in [0, \pi] \)
Step 2: Evaluate Each Term
We know:
\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\]
So, \[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]
So, \[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \]
Also,
\[
\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\]
So, \[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
So, \[ \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
Step 3: Add the Values
\[
-\frac{\pi}{3} + \frac{2\pi}{3}
= \frac{\pi}{3}
\]
Final Answer
\[
\frac{\pi}{3}
\]