evaluate inverse sine expression
Check the final answer first, then review the worked steps.
Problem
evaluate inverse sine expression
Answer
\(\frac{14\pi}{3}\)
Step-by-step solution
- Identify the inverse sine value: The expression involves $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$. The inverse sine function $\sin^{-1}(x)$ returns the angle $\theta$ in the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $\sin(\theta) = x$. We know from the unit circle that $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$. Therefore, $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$.
- Substitute the value into the expression: Now substitute $\frac{\pi}{3}$ back into the original expression: $-\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) + 5\pi = -\frac{\pi}{3} + 5\pi$.
- Simplify the expression: To combine the terms, find a common denominator, which is 3: $-\frac{\pi}{3} + \frac{15\pi}{3} = \frac{14\pi}{3}$.