The problem asks to evaluate the inverse sine function $\sin^{-1}(\sqrt{3}/2)$ for all appropriate values.

Solution:

1. Definition of $\sin^{-1}(x)$:

   • The inverse sine function, $\sin^{-1}(x)$, returns an angle $\theta$ such that $\sin(\theta) = x$.
   • The principal range for $\sin^{-1}$ is $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.

2. Identify the angle:

   • The value $\sin(\theta) = \sqrt{3}/2$ corresponds to $\theta = \frac{\pi}{3}$ in radians (or $60^\circ$) in the unit circle's first quadrant, because the sine of $\frac{\pi}{3}$ is $\sqrt{3}/2$.

3. All appropriate values:

   • The sine function is periodic with a period of $2\pi$, so the general solutions for $\sin(\theta) = \sqrt{3}/2$ occur at:
     • $\theta = \frac{\pi}{3} + 2n\pi$ (first quadrant)
     • $\theta = \pi - \frac{\pi}{3} + 2n\pi$ (second quadrant),
       where $n$ is any integer to account for the periodicity.

Final Answer:

The complete set of solutions for $\sin^{-1}(\sqrt{3}/2)$ is:
$\theta = \frac{\pi}{3} + 2n\pi \quad \text{and} \quad \theta = \frac{2\pi}{3} + 2n\pi, \quad \text{where } n \in \mathbb{Z}.$

Would you like a detailed explanation on any part or further clarifications?

5 Related Questions:

1. What is the principal value of $\sin^{-1}(\sqrt{3}/2)$?
2. How does the unit circle help find all possible solutions for $\sin^{-1}(x)$?
3. What are the general solutions for other sine values like $\sin^{-1}(1/2)$?
4. How does periodicity affect trigonometric solutions in general?
5. Can you explain the relationship between inverse functions and their principal ranges?

Tip:

When solving inverse trigonometric equations, always consider the principal range and the periodic nature of trigonometric functions for all solutions.