Find the minimum value of $5\sin \theta +12\cos \theta$.

An expression of the form $a\sin \theta +b\cos \theta$ has a minimum value of $-\sqrt{a^2+b^2}$ and a maximum value of $\sqrt{a^2+b^2}$.
Here, $a=5$ and $b=12$.

$$\text{Minimum value}=-\sqrt{5^2+12^2}=-\sqrt{25+144}=-\sqrt{169}=-13$$