Solving simultaneous equations

[Aelo]

Homework Statement

A uniaxial crystal of indices n₀ and n_e is cut so that the optic axis is
perpendicular to the surface. Show that for light from the outside at
an angle of incidence θ_i, the angle of refraction of the e-ray is

Homework Equations

The Attempt at a Solution

I solved for n_e by dividing both sides of the third equation by what appears to be an ellipse equation and taking the square root of both sides, then plugging it in into Snell's law (the second equation). It's become ugly and confusing, though. Help is appreciated.

 

[Mark44]

Homework Statement

A uniaxial crystal of indices n₀ and n_e is cut so that the optic axis is
perpendicular to the surface. Show that for light from the outside at
an angle of incidence θ_i, the angle of refraction of the e-ray is

Homework Equations

The Attempt at a Solution

I solved for n_e by dividing both sides of the third equation by what appears to be an ellipse equation and taking the square root of both sides, then plugging it in into Snell's law (the second equation). It's become ugly and confusing, though. Help is appreciated.

What is θ, that you have in your 2nd and 3rd equations (along with θ_i and θ_e)?

Your first equation should be using the lower-case forms of θ_i and θ_e, rather than the upper-case forms ($\Theta_i$ and $\Theta_e$). That could be confusing to some.

 

[Aelo]

Sorry about that... I screenshotted it straight from our PDF textbook. I believe θ = θ_e.

 

[Aelo]

I solved it, thanks anyway.