To solve the problem of evaluating [tex]\(\sin \left(\sin^{-1} \frac{\sqrt{3}}{2}\right)\)[/tex], let's go through the steps in detail:
1. Understand the inner function:
- The inner function is [tex]\(\sin^{-1} \frac{\sqrt{3}}{2}\)[/tex].
- [tex]\(\sin^{-1} x\)[/tex] (also known as [tex]\(\arcsin\)[/tex] or the inverse sine function) returns an angle whose sine value is [tex]\(x\)[/tex].
- So, [tex]\(\sin^{-1} \frac{\sqrt{3}}{2}\)[/tex] gives an angle [tex]\(\theta\)[/tex] such that [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex].
2. Identify the angle:
- Given that all angles are in Quadrant I (where all trigonometric values are positive), we recognize that [tex]\(\theta\)[/tex] must be the angle for which the sine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
- In Quadrant I, [tex]\(\theta = \frac{\pi}{3}\)[/tex] (or 60 degrees) satisfies [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex].
3. Apply the outer function:
- Now, we need to evaluate [tex]\(\sin(\theta)\)[/tex] where [tex]\(\theta = \sin^{-1} \frac{\sqrt{3}}{2}\)[/tex].
- Because by definition of [tex]\(\theta\)[/tex], [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex].
Therefore,
[tex]\[
\sin \left(\sin^{-1} \frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}
\][/tex]
From the given choices:
a. [tex]\(\frac{3}{2}\)[/tex]
b. [tex]\(\sqrt{3}\)[/tex]
c. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
d. [tex]\(\sqrt{2}\)[/tex]
The best answer is:
[tex]\[
\boxed{\frac{\sqrt{3}}{2}}
\][/tex]
