Why convert sine to cosine when deriving the law of reflection

[KaseyKC]

I'm trying to derive the law of reflection for Electromagnetic Waves and Optics. I'm using some lecture notes that my university provided. I'm confused as to why the two sine functions are changed to cosine functions as you don't do the same when you are deriving Snell's law.

Refer to attached for the equations.

I'm currently stuck at the part just before the sine's are converted into cosine functions.

 

[BvU]

Hello KKC, and welcome to PF !

You do want to use the template in the homework forum (it's mandatory, and the good spirits that watch over us get nasty if you don't).

However, for your question: I think it's a misprint in the book. They say $k_{ Iz} = k_{ Rz}$ but they work out $k_{ Ix} = k_{ Rx}$. The first -- as you correctly point out -- leads to $\sin \theta_I = \sin \theta_R \Rightarrow \theta_I = \theta_R$.

Personally, I don't really like the second ($k_{ Ix} = k_{ Rx}$), because $\cos \theta_I = \cos \theta_R \Rightarrow \theta_I = \theta_R$ isn't even correct (should be $\Rightarrow \theta_I = \pm \; \theta_R$).

I leave it to you to discover why the "other solution" of $\sin \theta_I = \sin \theta_R$ does not make it wrong to write $\Rightarrow \theta_I = \theta_R$ !​