How do I solve for the angle, as a function?

[Phantoful]

Homework Statement

Homework Equations

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The two equations in the image, as well as (maybe) standard vector operations like dot product, cross product.

The Attempt at a Solution

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So, I've tried to do what my instructor suggested and put everything in terms of Earth (Using R_E and T_E, but I'm not sure what to do from there as it doesn't seem that I'm progressing towards the solution. I've also tried to graph these functions, but I can't wrap my head around the fact that both the x and y vectors also have parameter t (time), so I'm not sure if graphing is the way to solve it. The main goal I've tried was to solve for theta, which doesn't seem possible because of the equations for x and y being split...

 

[haruspex]

Can you express the desired angle in terms of the two vectors, $\vec R_M, \vec R_E$? I.e., as a vector equation?

 

[Phantoful]

Can you express the desired angle in terms of the two vectors, $\vec R_M, \vec R_E$?

I would think I'd need to do the dot product using |A|*|B|cosθ and then calculate theta, but would it even be possible to find the magnitude? The vectors are changing over time and I don't know how to account for that.

Edit: Sorry, re-read your question. Would I have to convert these 'parametric equations' to an x-y equation, and then graph?

 

[haruspex]

the dot product using |A|*|B|cosθ

Dot product, yes, but not using that formula to find it. How do you perform a dot product on two vectors that are expressed in the same orthonormal basis?

 

[Phantoful]

Dot product, yes, but not using that formula to find it. How do you perform a dot product on two vectors that are expressed in the same orthonormal basis?

I can also add after multiplying the x's and the y's, but that would be a scalar, is that what I want? So I'm adding cos(2π(t/T_M))*cos(2π(t/T_E))+sin(2π(t/T_M))*sin(2π(t/T_E)), or is it the (u⋅v/v⋅v)v equation?

 

[haruspex]

I can also add after multiplying the x's and the y's, but that would be a scalar, is that what I want? So I'm adding cos(2π(t/T_M))*cos(2π(t/T_E))+sin(2π(t/T_M))*sin(2π(t/T_E)), or is it the (u⋅v/v⋅v)v equation?

Sorry, I think I've led you off in the wrong direction. Let's start again.

Draw a diagram of the Sun, Earth and Mars at some instant in the coordinate system given. Say we label the points S, E, M, and the distant star that fixes the X axis call P.
You can easily relate 2π(t/T_E) and 2π(t/T_M) to angles in the diagram, right?
What angle are you asked to find?

 

[Phantoful]

Sorry, I think I've led you off in the wrong direction. Let's start again.

Draw a diagram of the Sun, Earth and Mars at some instant in the coordinate system given. Say we label the points S, E, M, and the distant star that fixes the X axis call P.
You can easily relate 2π(t/T_E) and 2π(t/T_M) to angles in the diagram, right?
What angle are you asked to find?

I'm not actually sure of 'what' angle I need to find, which is my big problem, because I can't visualize this well so I'm depending on the mathematics. The question says "Derive a formula for which we see Mars from Earth. This should be an angle with respect to the

-axis." I can't easily relate the equations to the diagram, because I don't have a clue what these should look like.

 

[Phantoful]

This question will be the end of me...

 

[haruspex]

I'm not actually sure of 'what' angle I need to find, which is my big problem, because I can't visualize this well so I'm depending on the mathematics. The question says "Derive a formula for which we see Mars from Earth. This should be an angle with respect to the View attachment 219525-axis." I can't easily relate the equations to the diagram, because I don't have a clue what these should look like.

Did you draw the diagram as I described?
What we are after is the angle that the line from Earth to Mars makes to the x axis.
We have vector representations for the positions of Earth and Mars. What vector represents the position of Mars relative to Earth?