Let $x = A\sin(at+\theta)$ and $y = A\sin(at)$ . Prove that this parametric equation forms an ellipse except when $\theta = 0 $ and $\pi$ .
My try : I expanded the $\sin(at + \theta)$ and looked for a useful relation between $x$ and $y$ but didn't work . I think it should be a rotated ellipse and the rotation matrix is involved .
Best Answer
Let $t=u-\dfrac\theta{2a}$. Then $x=A\sin\left(au+\dfrac{\theta}{2}\right)$ and $y=A\sin\left(au-\dfrac{\theta}{2}\right)$.
$x+y=2A\sin au\cos\dfrac\theta2$ and $x-y=2A\cos au\sin\dfrac\theta2$.
$\displaystyle \left(\frac{x+y}{2A\cos\frac{\theta}{2}}\right)^2+\left(\frac{x-y}{2A\sin\frac{\theta}{2}}\right)^2=1$
It is an ellipse with axes making $45^\circ$ with the coordinate axes.