Why do we only use sine in snell's law why not cos or tan?

Question

Why do we only use sine in snell's law why not cos or tan?

Answer 1

Because we choose to measure the angles from the normal.  If you measured them from the boundary, it would be cosine.  If you want to actually derive Snell's law, you have to understand a little about electricity and magnetism.  At a boundary, you have to match certain components of the EM fields of the incoming and outgoing rays, and that determines Snell's law (and the law of reflection and total internal reflection and evanescent waves and all that stuff).

You can also derive it from simpler assumptions.  One is Fermat's principle--that light takes the path to its destination that is a minimum (or an extreme value) of time.  And that fastest path turns out to be the bent lines defined by Snell's law.  Fermat's principle turns out to be more than just a mathematical trick, but actually has a deeper significance in terms of mechanics.

The link Greydog gave makes a different (and less obvious) assumption--that two pulses of light on parallel paths enter the boundary next to one another and exit next to one another--niether gets ahead

"If d1 and d2 represent the distances traveled in the respective mediums during the same amount of time...."

I don't see any obvious physical motivation for that, but it seems to get the right answer, so it must be equivalent.

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Pearl:

"Since the distances are proportional to the sines of the angles"

Why?

Answer 2

You may rewrite the law in terms of ANY trigonometric function you like, but the "easiest" in terms of expressing the law is "sine".  When deriving the amount of bend that the wavefront of light encounters when entering a more optically dense substance, "Sine" is the most appropriate choice for brevity.

Here is the derivation, if you care to see why...

http://dev.physicslab.org/Document.aspx?...

Answer 3

we cannot use tan function because it is discontinous at odd multiple of pi/2 while sin cos are continous function and they are inter convertable

Answer 4

Sine formula is the best preffered formula for calculating refracting index if you use cos or tan you can't get a perfect and accurate result if you use tan or cos so using sine gives most accurate results  

Answer 5

By definition, the refractive index of a medium is the ratio of the speed of light in vacuum to the speed of light in a medium.

μ = c / v

If, μ 1 and μ2 are the respective indices of refraction and v1 and v2 are the speeds of light in the medium 1 and medium 2 , then

μ 1/ μ2 = v2 /v1

For the same interval of time this can be written as

μ 1/ μ2 = d2 / d1  where d2 and d1 are the distances traveled in a given interval of time in the second and fist medium respectively.

Now the problem is to find the relation between the distance and angles.

Since the distances are proportional to the sines of the angles it follows   that

μ 1/ μ2 = sin i1  / sin r2 .

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If we measure the angle from the inter face [which should be perpendicular the normal] then the above formula should be modified as

μ 1/ μ2 =cos i1’ / cos r2’ where the angles are measured from the interface.

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