# Snell's Law Variant

1. Sep 28, 2008

### Cider

Is there a way to express Snell's Law using cosines of the angles of incidence instead of the sines without the cosines being squared? If no one here knows, is there anywhere I could look into this question?

2. Sep 28, 2008

### cepheid

Staff Emeritus
Note that

$$\sin \theta =\cos(\theta - \frac{\pi}{2} )$$

Also,

cos(-x) = cos(x), therefore:

$$\cos(\theta - \frac{\pi}{2} ) = \cos( \frac{\pi}{2} - \theta)$$

pi/2 - theta = 90 degrees - theta = the angle the ray makes with the *surface* (instead of with the normal)

So IF you use angles of incidence and reflection defined as the angles the rays make with the surface instead of the angles they make with the normal to the surface, THEN Snell's law would indeed be expressed in terms of the cosines of THOSE angles. However, I would not encourage you to do this, because that is not the conventional definition for angles of incidence and reflection in optics. If you use that definition without telling somebody, and claim the angle of incidence is 35 degrees, he will think you are talking about the angle wrt the normal, which would actually be 55 degrees in that case.

3. Sep 28, 2008

### Cider

Sorry, I probably should have made myself more clear. I need the equation to use only $$\cos \theta$$ with no subtractions or additions within the cosine. And $$\theta$$ has to be the angle of incidence. It cannot be the compliment to that angle.