# Thread: find the parametric equation of an ellipse?

1. ## find the parametric equation of an ellipse?

how to find the parametric equation of an ellipse?

x^2/a^2 + y^2/b^2 =1

thank you very much.

2. Originally Posted by kittycat
how to find the parametric equation of an ellipse?

x^2/a^2 + y^2/b^2 =1

thank you very much.
the ellipse $\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$ is given by the parametric equations: $x = a \cos t \mbox{ , } y = b \sin t$ for $0 \le t \le 2 \pi$

3. hi Jhevon,

How do you work out this parametric equation?

please teach me . thank you.

4. Originally Posted by kittycat
hi Jhevon,

How do you work out this parametric equation?

please teach me . thank you.
we know that $\cos^2 t + \sin^2 t = 1$

and we also know from our knowledge of polar coordinates that it is more appropriate to relate the x to the cosine and the y to the sine, so

$\frac {x^2}{a^2} + \frac {y^2}{b^2} = \left( \frac xa \right)^2 + \left( \frac yb \right)^2 = 1$

thus we can let $\cos t = \frac xa$ and $\sin t = \frac yb$ (and so we would obtain $\cos^2 t + \sin^2 t = 1$)

solving for $x$ and $y$ we get: $x = a \cos t$ and $y = b \sin t$ and we restrict $t$ to $0 \le t \le 2 \pi$ since sine and cosine are periodic for that interval

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# how to find parametric equation of ellipse

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