Let $a, b \in \mathbf{R}^{*}$, find the minimum value of $y=\frac{a}{\sin ^{3 / 2} \theta}+\frac{b}{\cos ^{3 / 2} \theta}\left(\theta \in\left(0, \frac{\pi}{2}\right)\right)$.
Introducing an undetermined positive constant $k$, we have
$$
\begin{aligned}
y & =4 \cdot \frac{a}{4 \sin ^{3 / 2} \theta}+3 k \sin ^{2} \theta+4 \cdot \frac{b}{4 \cos ^{3 / 2} \theta}+3 k \cos ^{2} \theta-3 k \\
& \geqslant \sqrt[7]{\left(\frac{a}{4 \sin ^{3 / 2} \theta}\right)^{4}\left(k \sin ^{2} \theta\right)^{3}}+\sqrt[7]{\left(\frac{b}{4 \cos ^{3 / 2} \theta}\right)^{4}\left(k \cos ^{2} \theta\right)^{3}}-3 k \\
& =7 \sqrt[7]{\left(\frac{a}{4}\right)^{4} k^{3}}+7 \sqrt[7]{\left(\frac{b}{4}\right)^{4} k^{3}}-3 k .
\end{aligned}
$$

Equality holds if and only if $\frac{a}{\sin ^{3 / 2} \theta}=k \sin ^{2} \theta$ and $\frac{b}{\cos ^{3 / 2} \theta}=k \cos ^{2} \theta$,
i.e., $\tan ^{7 / 2} \theta=\frac{a}{b} \Leftrightarrow \tan \theta=\left(\frac{a}{b}\right)^{2 / 7}$,
i.e., $\theta=\arctan \left(\frac{a}{b}\right)^{2 / 7}$.
Thus, when $\theta=\arctan \left(\frac{a}{b}\right)^{\frac{2}{7}}$, $y$ attains its minimum value, which is easily found to be $\left(a^{\frac{4}{7}}+b^{\frac{4}{7}}\right)^{\frac{7}{4}}$.