what exactly happens when I increase the sampling period ?
Providing some Matlab plots so that I can be more clear.
Using the signal $x(t) = \sin(10\pi t)$ and periods $T_s=0.02 ,\; 0.05 ,\; 0.1$
What happens exactly to the signal when the $T_s = 0.1s$ ?
You sample at times $nT_s$, so the values of the sampled signal are
$$x_d[n]=x(nT_s)=\sin(10\pi nT_s)\big |_{T_s=0.1}=\sin(n\pi)=?$$
Can you take it from here?
You are sampling a sine function that should return zero, but because of finite precision argument into that sine function, the error tends to increase towards the right?
Your x-axis is apparently varying between plots. And Matlab plots joining lines between discrete points are misleading. So, here is an other version, with more sampling times:
The Nyquist-etc sampling theorem tells you (somehow) that if you don't have at least two samples per period, you are likely to loose most of the signal's information. Until $0.095$ sampling, although you seem to sometimes loose on dynamics, you still have theoretical hope (if the discrete signal were infinite, etc.) that you can recover the continuous data. Yet, as the sampling time increases, it become more difficult to imagine the red signal from the black crosses. And indeed, as time sampling reaches $1/(2*5)$ (the $5$ as in $\sin(2 \pi 5 t)$), almost all is lost, as you can see from the unscaled graphs, very close to zero for the last one:
