- #1
McLaren Rulez
- 292
- 3
Homework Statement
Prove that the equations [itex]x=acos(\theta)[/itex] and [itex]y=bcos(\theta +\delta )[/itex] is the equation of an ellipse and what angle does this ellipse's major axis make with the x axis?
Homework Equations
Equation of an ellipse is [itex]x=acos\theta, y=asin\theta[/itex]
Rotation matrix is for a rotation by [itex]\psi[/itex] is [tex]A=\begin{pmatrix} cos\psi & -sin\psi \\ sin\psi & cos\psi \end{pmatrix}[/tex]
The Attempt at a Solution
I know the special case of [itex]\delta = \pi/2[/itex] is easy but I cannot do it for arbitrary [itex]\delta[/itex]. I worked out what an ellipse whose major axis forms an angle [itex]\psi[/itex] with the x-axis looks like. I did this by applying a rotation matrix to the standard equation [itex]x=acos\theta, y=bsin\theta[/itex]
This gives [itex]x= acos\theta cos\psi - bsin\theta sin\psi[/itex] and [itex]y= acos\theta sin\psi + bsin\theta cos\psi [/itex]
Now, what is the relation between [itex]\psi[/itex] and [itex]\delta[/itex] in general. And I need to show that the equations [itex]x=acos(\theta)[/itex] and [itex]y=bcos(\theta +\delta )[/itex] can be brought to the same form as [itex]x= acos\theta cos\psi - bsin\theta sin\psi[/itex] and [itex]y= acos\theta sin\psi + bsin\theta cos\psi [/itex]