Determining Sinusoid Parameters

$\begin{aligned}y(t)&=A\sin (\omega t+\theta)\\ y(1)&=A\sin (\omega +\theta)\approx 2.32705644463\\ y(2)&=A\sin (2\omega +\theta)\approx -2.4785144342\\ y(3)&=A\sin (3\omega +\theta)\approx -3.10007384142\\ y(3)+y(1)&=A\sin (3\omega +\theta)+A\sin (\omega +\theta)\\ &=2A\sin (2\omega +\theta)\cdot \cos(\omega)\hspace{5mm}\color{#3D99F6}\small\sin A+\sin B=2 \sin\left(\dfrac{A+B}{2}\right)\cos\left(\dfrac{A-B}{2}\right)\\ &=2y(2)\cos (\omega)\\ \implies\cos(\omega)&=\dfrac{y(3)+y(1)}{2y(2)}\\ \cos(\omega)&\approx\dfrac{(-3.10007384142)+(2.32705644463)}{2(-2.4785144342)}\\ \cos(\omega)&\approx0.15594369476397\\ \sin(\omega)&\approx\pm0.98776594599295\\\\ y(2)&=A\sin (2\omega +\theta)\\ &=A\sin ((\omega +\theta)+\omega)\\ &= A\sin (\omega +\theta)\cos(\omega)+A\cos (\omega +\theta)\sin(\omega)\\ &=y(1)\cos(\omega)+A\cos (\omega +\theta)\sin(\omega)\\ A\cos (\omega +\theta)&=\dfrac{y(2)-y(1)\cos(\omega)}{\sin(\omega)}\\ &\approx=\dfrac{(-2.4785144342)-(2.32705644463)(0.15594369476397)}{\pm0.98776594599295}\\ &\approx\mp 2.8765966529\\\\ A&=\sqrt{(A\cos (\omega +\theta))^2+(A\sin (\omega +\theta))^2}\\ &\approx\sqrt{(\mp2.8765966529)^2+(2.32705644463)^2}\\ &\approx{3.700000000_{\cdots}}\\\\ \implies A&\approx\color{#EC7300}\boxed{\color{#333333}3.7}\end{aligned}$

One can use a curve fit program and get 3.7 as well.