Solution: $2 \sin \theta+3 \cos \theta =2\left(\frac{2 \tan \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right)+3\left(\frac{1-\tan ^{2} \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}\right)$

$=2\left(\frac{2 \times \frac{2}{3}}{1+\left(\frac{2}{3}\right)^{2}}\right)+3\left(\frac{1-\left(\frac{2}{3}\right)^{2}}{1+\left(\frac{2}{3}\right)^{2}}\right) =3$