Four, (20 points) Given that $a$ and $b$ are 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 $\sin \theta$.
Let $x=\cos \theta, y=\sin \theta$, then from the given conditions we have
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=1, \\
b x+a y \geqslant 0, \\
a x-b y \geqslant 0 .
\end{array}\right.
$$

where $|x| \leqslant 1,|y| \leqslant 1$.
When equations (2) and (3) are equal, the lines
$$
b x+a y=0
$$

and $a x-b y=0$
are perpendicular to each other. The problem is then transformed into "finding the maximum y-coordinate of the arc $C=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}$ within the region
$$
G=\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}
b x+a y \geqslant 0, \\
a x-b y \geqslant 0,
\end{array}|x| \leqslant 1,|y| \leqslant 1\right\}\right.\right.
$$

Based on the signs of $a$ and $b$, we discuss 4 cases.
(1) $a>0, b>0$.
As shown in Figure 8, we need to find the y-coordinate of the intersection point $P_{1}$ of line (5) and circle (1).

The result is $y_{\max }=y_{P_{1}}=\frac{a}{\sqrt{a^{2}+b^{2}}}$.
(2) $a>0, b<0$.
As shown in Figure 11, the y-coordinate of the intersection point $P_{4}$ of line (4) and circle (1) is $y_{P_{4}}=$ $\frac{-b}{\sqrt{a^{2}+b^{2}}}$; the y-coordinate of the intersection point $P_{5}$ of line (5) and circle (1) is $y_{P_{5}}=$ $\frac{a}{\sqrt{a^{2}+b^{2}}}$.
Comparing $y_{P_{4}}$ and $y_{P_{5}}$, we find that
when $a \leqslant -b$, $y_{\text {max }}=y_{P_{4}}=\frac{-b}{\sqrt{a^{2}+b^{2}}}$;
when $a > -b$, $y_{\max }=y_{P_{5}}=\frac{a}{\sqrt{a^{2}+b^{2}}}$.