Question 5 As shown in Figure 6, find the maximum area of an arbitrary quadrilateral $ABCD$ inscribed in a semicircle $O$ with radius 1.
Parse: Connect $O C, O D$, let
$$\angle B O C=\alpha, \angle C O D=\beta, \angle D O A=\gamma$$

Obviously $\alpha+\beta+\gamma=\pi$, then the area of quadrilateral $A B C D$
$$S=\frac{1}{2}(\sin \alpha+\sin \beta+\sin \gamma)$$

By the common inequality $\sin \alpha+\sin \beta+\sin \gamma \leqslant \frac{3 \sqrt{3}}{2}$, we immediately know $S_{\max }=\frac{3 \sqrt{3}}{4}$.