How to estimate amplitude of a sinusoid from two samples

Question

Given a sinusoid $x(t)=A\cos(t+\theta)$, I can estimate the amplitude $A$ if I take two samples separated by $\frac{\pi}{2}$. If $X_1=x(0+\theta)$ and $X_2=x(\pi/2+\theta)$. Then, $A$ can be estimated from $X_1^2+X_2^2$.

However, I am taking samples separated by a different (but known) angle $\phi$. The samples are

$X_1=x(0+\theta)$ and $X_2=x(\phi+\theta)$

Can I estimate $A$ from these two sample?

Answer 1

$$X_1=A\cos{\theta}$$ $$X_2=A\cos{(\theta+\phi)}=A\cos\theta cos\phi-A\sin{\theta}\sin{\phi}$$

Therefore:

$$X_2=X_1\cos{\phi}-A\sin{\phi}\sqrt{1-\frac{X_1^2}{A^2}}$$ $$\sqrt{A^2-X_1^2}=\frac{X_1\cos{\phi}-X_2}{\sin{\phi}}$$ $$A^2=X_1^2+\left(\frac{X_1\cos{\phi}-X_2}{\sin{\phi}}\right)^2$$