Example 22 Skew lines $a, b, a \perp b, c$ forms a $30^{\circ}$ angle with $a$, find the range of the angle $\theta$ formed between $c$ and $b$.
Decompose the construction of a cone as shown in Figure $4-3$: the slant height is $c$, the height is $a$, and $c$ forms a $30^{\circ}$ angle with $a$. The axial section $P A B$ $\perp$ the axial section $P C D$, and $A B / / b$, which implies $C D \perp A B, A B \perp$ the axial section $P C D$. Therefore, the angle $\theta$ formed by $c$ and $b$ is the angle formed by the slant height $c$ and a line parallel to $A B$.

When $c$ is in the position of $P A$, since $\angle O P A=30^{\circ}$, then $\angle P A O=\theta=50^{\circ}$, at this time $\sin \angle P A O=$ $\frac{a}{c}$.

When $c$ rotates clockwise from the position of $P A$ to $P M$, draw $M E / / A B / / b$, intersecting $C D$ at $E$, we get $M E \perp C D, \angle P M E=\theta, \sin \angle P M E=\frac{P E}{c}>\frac{P()}{c}=\frac{a}{c}=\sin \angle P A O$.
Thus, $\angle P M E>\angle P A O=60^{\circ}$, i.e., $\theta>60^{\circ}$.
When $c$ is in the position of $P C$, $C O \perp A B$, by the theorem of three perpendiculars, we get $c \perp \alpha, \theta=90^{\circ}$, hence the range of the angle $\theta$ formed by $c$ and $b$ is $\left[60^{\circ}, 90^{\circ}\right]$.