On $\left(0,\frac\pi2\right)$ all terms are positive. On $\left(\frac\pi2,\pi\right]$, $\sin x+2\tan x=\sin x\left(1+\frac2{\cos x}\right)\le0$ (as $\frac2{\cos x}\le-2$) and $3\cos x<0$, so $f<0$. No roots there.
On $\left(\pi,\frac{3\pi}2\right)$, $f'(x)=\cos x-3\sin x+\frac2{\cos^2x}\ge-1+0+2>0$, so $f$ is increasing and has at most one root. Now $f(4)\approx-0.757-1.961+2.316<0$ and $f(4.5)\approx-0.978-0.633+9.27>0$, so the root lies in $(4,4.5)$.
The smallest positive root lies in $\boxed{(4,5)}$.
