Snell's law and inverse function

[Karol]

Homework Statement

Snell's law is:
$$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$
$$\frac{c_1}{c_2}=n_{12}$$
Express ##\theta_2## as a function of ##\theta_1##
Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is defined (for larger values of ##\theta_1## than this the incoming light will be reflected).

Homework Equations

Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x##

The Attempt at a Solution

$$\sin ( \theta_2 )=\frac{\sin( \theta_1 ) }{n_{12}}~\rightarrow~\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\theta_2$$
##\theta_2## can be ##\frac{\pi}{2}## at the max:
$$\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\frac{\pi}{2}~~\rightarrow~~\sin\left( \frac{\pi}{2} \right)=\frac{\sin(\theta_1)}{n_{12}}$$
$$\Rightarrow~\sin(\theta_1)=n_{12}\sin\left( \frac{\pi}{2} \right)=n_{12}$$
$$\theta_1<\arcsin(n_{12})$$
I didn't use at all the definition of inverse function in the second question, i feel what i have done isn't what it's meant from the chapter of inverse functions

 

[haruspex]

I didn't use at all the definition of inverse function in the second question

Not sure what you mean by that. The answer you gave involved an inverse function, and your final step involved inverting the sine function. Can you be more specific about what it is that you feel you have not used?
By the way, the diagram doesn't match the text. It clearly shows a case with c₂>c₁, whereas the text assumes the reverse.

 

[Karol]

the diagram doesn't match the text. It clearly shows a case with c2>c1, whereas the text assumes the reverse.

How does the text show anything about the diagram? only the last formula: ##\theta_1<\arcsin(n_{12})## to my opinion, may show something in that direction.
if ##c_2>c_1~~\rightarrow~\sin(\theta_2)>\sin(\theta_2)~\rightarrow~\theta_2>\theta_1## and the diagram shows the inverse.
The diagram for ##c_2>c_1## must be:

for ##c_2>c_1~\rightarrow~n_{12}<1## and ##\theta_1## has a definite value and that's correct.

 

[haruspex]

How does the text show anything about the diagram?

I confused you by incorrectly describing the mismatch.
The diagram shows an example of c₁>c₂, as you say. But in that case there is no limit (before π/2) on θ₁. n₁₂>0, so the arcsin of it is not defined.
Your second diagram, with c₂>c₁, makes more sense. E.g. total internal reflection can occur from water (low c) to air (high c), but not the other way around.

 

[Karol]

The diagram shows an example of c₁>c₂, as you say. But in that case there is no limit (before π/2) on θ₁. n₁₂>0, so the arcsin of it is not defined.
Your second diagram, with c₂>c₁, makes more sense. E.g. total internal reflection can occur from water (low c) to air (high c), but not the other way around.

True, that is also why i thought my answer wasn't right, since there is no limit for ##\theta_1## if ##n_{12}>1##