Finding the Major Axis Angle of an Ellipse Given Rotated Equations

[McLaren Rulez]

Homework Statement

Prove that the equations $x=acos(\theta)$ and $y=bcos(\theta +\delta )$ 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 $x=acos\theta, y=asin\theta$
Rotation matrix is for a rotation by $\psi$ is $$A=\begin{pmatrix} cos\psi & -sin\psi \\ sin\psi & cos\psi \end{pmatrix}$$

The Attempt at a Solution

I know the special case of $\delta = \pi/2$ is easy but I cannot do it for arbitrary $\delta$. I worked out what an ellipse whose major axis forms an angle $\psi$ with the x-axis looks like. I did this by applying a rotation matrix to the standard equation $x=acos\theta, y=bsin\theta$

This gives $x= acos\theta cos\psi - bsin\theta sin\psi$ and $y= acos\theta sin\psi + bsin\theta cos\psi$

Now, what is the relation between $\psi$ and $\delta$ in general. And I need to show that the equations $x=acos(\theta)$ and $y=bcos(\theta +\delta )$ can be brought to the same form as $x= acos\theta cos\psi - bsin\theta sin\psi$ and $y= acos\theta sin\psi + bsin\theta cos\psi$

 

[vela]

This might not be the most efficient method. Use
$$\cos \theta = \frac{x}{a}$$ to eliminate $\theta$ from
$$\frac{y}{b} = \cos(\theta+\delta) = \cos \theta \cos \delta - \sin\theta \sin \delta$$ to get the equation of an ellipse in standard form.

 

[McLaren Rulez]

Thanks Vela. It took some doing but it worked!