Understanding the Ellipse Equation: Cartesian or Polar Coordinates?

[lionelwang]

Hi, guys,

Is the ellipse equation "x=acost; y=bcost" a Cartesian coordinates equation or a polar coordinates equation? Someone said that it's a transfer from a polar one to a Cartesian one.
Need more help on this, thank you very much!

 

[bm0p700f]

Cartesian I think as you get x and y-axis coordinates.

 

[lionelwang]

Thanks!

 

[micromass]

Is the ellipse equation "x=acost; y=bcost"

You did mean $x=a\cos(t),~~y=b\sin(t)$, right?

 

[HallsofIvy]

These are parametric equations giving Cartesian coordinates. If you really mean x= a cos(t), y= b cos(t), you can solve the first equation as cos(t)= x/a so the second equation becomes y= (b/a)x, which graphs as a straight line.

If you meant x= a cos(t), y= b sin(t), as micromass suggests, then x/a= cos(t), y/b= sin(t) so that $(x/a)^2+ (y/b)^2= cos^2(t)+ sin^2(t)= 1$, an ellipse.

The equations relating polar coordinates and Cartesian coordinates are different but similar: $x= r cos(\theta)$, $y= r sin(\theta)$.

 

[lionelwang]

Thank you very much, guys.