Proof of Elliptical Motion through R(t) Equation

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The discussion centers on proving that the position vector r(t) = A cos(ωt) i + B sin(ωt + α) describes elliptical motion. The original poster recognizes the standard parametric equations for an ellipse but struggles to find a relevant trigonometric identity for their specific equation. Participants suggest that the general parametric equation for an ellipse can demonstrate that the given equation fits an elliptical path, despite its rotation. The challenge remains in providing a clear proof of this elliptical motion. The conversation emphasizes the need for understanding how to manipulate trigonometric identities to establish the proof.
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Hi!, can anybody help me with this problem?

Proof that the position vector r(t) = A cos \omegat i + B sin (\omegat + \alpha) moves on an ellipse.

I understand that the parametric equation for an ellipse is x=a cos t & y=b sin t, but I just can't find any trigonometric identity that helps me. Do you know any math trick that could help me?
 
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x=a cos t, y=b sin t is only for a canonical ellipse. If you find the general parametric equation for an ellipse, you'll see that your equation fits perfectly.
 
I understood what you say, but the only thing I could see was that the ellipse is rotated, but I still don't know how to prove the statement
 

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