Three. (20 points) Given that $a$ and $b$ are real constants not equal to 0, and the variable $\theta$ satisfies the inequality system
$$
\left\{\begin{array}{l}
a \sin \theta + b \cos \theta \geqslant 0, \\
a \cos \theta - b \sin \theta \geqslant 0 .
\end{array}\right.
$$

Try to find the maximum value of $\sin \theta$.
From
\[
\left\{\begin{array}{l}
a \sin \theta + b \cos \theta \geqslant 0, \\
a \cos \theta - b \sin \theta \geqslant 0,
\end{array}\right.
\]
we know
\[
\left\{\begin{array}{l}
\sin (\theta + \varphi) \geqslant 0 \\
\cos (\theta + \varphi) \geqslant 0
\end{array}\right.
\]

where, \(\sin \varphi = \frac{b}{\sqrt{a^2 + b^2}}, \cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}\).
Thus, \(2 k \pi \leqslant \theta + \varphi \leqslant 2 k \pi + \frac{\pi}{2}\).
Therefore, \(2 k \pi - \varphi \leqslant \theta \leqslant 2 k \pi + \frac{\pi}{2} - \varphi\).

(1) When \(a > 0, b > 0\), \(\varphi\) is an acute angle.
The function \(\sin \theta\) is increasing on \(\left[2 k \pi - \varphi, 2 k \pi + \frac{\pi}{2} - \varphi\right]\), thus,
\[
\begin{array}{l}
(\sin \theta)_{\max} = \sin \left(2 k \pi + \frac{\pi}{2} - \varphi\right) \\
= \cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}.
\end{array}
\]

(2) When \(a > 0, b < 0\), \(\frac{\pi}{2} < \varphi < \pi\).
If \(b > -a\), then
\[
\begin{array}{l}
\sin (2 k \pi - \varphi) = -\sin \varphi = -\frac{b}{\sqrt{a^2 + b^2}} \\
> \sin \left(2 k \pi + \frac{\pi}{2} - \varphi\right) = \cos \varphi = \frac{a}{\sqrt{a^2 + b^2}}. \\
\text{Hence, } (\sin \theta)_{\max} = \sin (2 k \pi - \varphi) \\
= -\frac{b}{\sqrt{a^2 + b^2}}.
\end{array}
\]

Similarly, if \(-a \leqslant b\), then
\[
(\sin \theta)_{\max} = \sin \left(2 k \pi + \frac{\pi}{2} - \varphi\right) = \frac{a}{\sqrt{a^2 + b^2}}.
\]

(4) When \(a < 0, b < 0\), \(-\pi < \varphi < -\frac{\pi}{2}\). Since the function \(\sin \theta\) is decreasing on \(\left[2 k \pi - \varphi, 2 k \pi + \frac{\pi}{2} - \varphi\right]\), thus,
\[
\begin{array}{l}
(\sin \theta)_{\max} = \sin (2 k \pi - \varphi) = -\sin \varphi \\
= -\frac{b}{\sqrt{a^2 + b^2}}.
\end{array}
\]