If we know the values of a sine wave at $t_0$ and $t_1$, which we denote by $d_0$, is it possible to calculate the amplitude of that particular sine wave? We know only the difference $t_1 - t_0$, not the exact values of $t_0$ or $t_1$.
I am trying to find the amplitude of a sinusoidal vibration in a situation where the sensor can measure only the time difference $ t_1-t_0$ between consecutive threshold level of the vertical displacement $ d_0 $ being detected.
Here are my calculations. I get stuck when I get the $t_0 + t_1$ due to the fact I know only the difference $t_1 - t_0$.
$$ d_0 = A \sin (\omega t_0) \tag{1}$$
$$ d_0 = A \sin (\omega t_1) \tag{2}$$
Adding $(1)$ and $(2)$,
$$ 2 d_0 = A ( \sin (\omega t_0) + \sin (\omega t_1) )$$
and, thus,
$$ A = \frac{2d_0}{\sin(\omega t_0) + \sin(\omega t_1)} = \frac{d_0}{\sin\left(\omega \left(\dfrac{t_0 + t_1}{2}\right) \right) \cos\left(\omega\left(\dfrac{t_1 - t_0}{2}\right)\right)} $$
