Completeness of $sin(kx)_{k=0}^{\infty}$ in $L_1[1, 4]$

I am reading a function analysis book and having trouble proving the following task: is ${\sin(kx)}_{k=0}^{\infty}$ complete in $L_1[1, 4]$?

I understand that completeness in $C[1, 4]$ implies completeness in $L_1[1, 4]$, but I'm having trouble with this task because I can not use the following theorem (which is usually used to prove such facts):

1. Trigonometric system $\frac{1}{2}, \cos(kx), \sin(kx)$ is complete in a space of continuous 2-pi periodic functions

The problem is that $4 > \pi$, and I'm confused about how to handle this. Any help would be highly appreciated.

UPD: what if there would be system ${\sin(kx)}_{k=1}^{\infty}$, would it change something?