Example 6 Given $7 \sin \alpha + 24 \cos \alpha = 25$, find the value of $\sin \alpha$.

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Example 6 Given $7 \sin \alpha + 24 \cos \alpha = 25$, find the value of $\sin \alpha$.
Analysis: Observing that $7.24$ and $25$ form a Pythagorean triple, we square both sides of $7 \sin \alpha + 24 \cos \alpha = 25$, obtaining $25^{2} = (7 \sin \alpha + 24 \cos \alpha)^{2} \leqslant (\sin ^{2} \alpha + \cos ^{2} \alpha)(7^{2} + 24^{2}) = 25^{2}$. According to the condition for equality in the Cauchy-Schwarz inequality, we get $\frac{\sin \alpha}{7} = \frac{\cos \alpha}{24}$, from which we solve $\sin \alpha = \frac{7}{25}$.