Question

Evaluate: \(\frac{3\pi}{2} - \sin^{-1} \left( -\frac{1}{2} \right)\)

• A. \(0\)
• B. \(\frac{3\pi}{2}\)
• C. \(2\pi\)
• D. \(\pi\)

Hint:

For \(\sin^{-1} \left( -\frac{1}{2} \right)\), use the fact that \(\sin^{-1} \left( -\frac{1}{2} \right) = -\frac{\pi}{6}\) based on the unit circle.

Answer 1

The Correct Option is D

Step 1: Recall that \(\sin^{-1} \left( -\frac{1}{2} \right)\) corresponds to an angle \(\theta\) such that \(\sin \theta = -\frac{1}{2}\) and \(\theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]\). From the unit circle, we know: \[ \sin^{-1} \left( -\frac{1}{2} \right) = -\frac{\pi}{6} \] Step 2: Now evaluate: \[ \frac{3\pi}{2} - \left( -\frac{\pi}{6} \right) = \frac{3\pi}{2} + \frac{\pi}{6} \] Step 3: Simplify the expression: \[ \frac{3\pi}{2} + \frac{\pi}{6} = \frac{9\pi}{6} + \frac{\pi}{6} = \frac{10\pi}{6} = \pi \] Thus, the answer is \(\pi\). ---