Reconstructing a sine wave from an interval shorter than half its wavelength
If your signal is really as simple as
$$x(t)=A\sin(\omega_0t)\tag{1}$$
with known $\omega_0$, and you have observations $y(t_i)$, which are noisy samples of $x(t)$ at known time instances $t_i$, then a simple solution would be the least squares estimate
$$\hat{A}=\frac{\displaystyle\sum_iy(t_i)\sin(\omega_0t_i)}{\displaystyle\sum_i\sin^2(\omega_0t_i)}\tag{2}$$
Of course, this simple solution won't work if your signal actually has the form
$$x(t)=A\sin(\omega_0t+\phi)+c \tag{3}$$
with unknown phase $\phi$ and DC-offset $c$. However, you can compute optimum least squares estimates for that problem too. This is discussed in Cedron's answer.