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Solution Steps

The problem asks to find the exact value of the expression \( \sin(\arcsin(\frac{\sqrt{3}}{2})) \).

The inner function is \( \arcsin(\frac{\sqrt{3}}{2}) \). The arcsine function, \( \arcsin(x) \), returns the angle whose sine is \( x \). Therefore, \( \arcsin(\frac{\sqrt{3}}{2}) \) is the angle \( \theta \) such that \( \sin(\theta) = \frac{\sqrt{3}}{2} \).

The angle \( \theta \) for which \( \sin(\theta) = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \) (or 60 degrees), since \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).

Now, we need to find \( \sin(\theta) \) where \( \theta = \arcsin(\frac{\sqrt{3}}{2}) \). Since \( \theta = \frac{\pi}{3} \), we have: \[ \sin(\arcsin(\frac{\sqrt{3}}{2})) = \sin(\frac{\pi}{3}) \]

We know that \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).

Final Answer