Example 3 Given $\sin \varphi \cdot \cos \varphi=\frac{60}{169}$, and $\frac{\pi}{4}<\varphi$ $<\frac{\pi}{2}$. Find the values of $\sin \varphi$ and $\cos \varphi$.
Solution: Since $\sin \varphi \cdot \cos \varphi=\left(\frac{2}{13} \sqrt{15}\right)^{2}$, then $\sin \varphi$, $\frac{2}{13} \sqrt{15}$, and $\cos \varphi$ form a geometric sequence. Therefore, let
$$
\sin \varphi=\frac{2}{13} \sqrt{15} q, \cos \varphi=\frac{2}{13 q} \sqrt{15}(1<q
$$
$<2$ ).
Substituting into $\sin ^{2} \varphi+\cos ^{2} \varphi=1$, we get
$$
\left(\frac{2}{13} \sqrt{15} q\right)^{2}+\left(\frac{2}{13 q} \sqrt{15}\right)^{2}=1 \text {. }
$$

Simplifying and factoring, we get
$$
\left(12 q^{2}-5\right)\left(5 q^{2}-12\right)=0 \text {. }
$$
$\therefore q^{2}=\frac{12}{5}, q^{2}=\frac{5}{12}$ (discard). Hence $q=\frac{6}{\sqrt{15}}$.
$$
\therefore \sin \varphi=\frac{12}{13}, \cos \varphi=\frac{5}{13} \text {. }
$$