Is the Critical Angle Equal to the Brewster Angle?

[fredrick08]

Homework Statement

when the critical angle=brewster angle, what is the refractive index, if incident n is air=1.

arcsin(1/n)=arctan(n)=>n=1.272...?

how is this? i tried differentiating both side and solving but i just get into a big mess, can anyone help me if i am missing something?

 

[fredrick08]

ive tried pythagoras, and trig idendenties, but all i come up with is x=x??

 

[Cyosis]

If you want to use identities you can take the sin on both sides and then use $\cos^2x+\sin^2x=1,\;1+\tan^2x=\sec^2x$ to simplify the right hand side.

 

[fredrick08]

ok thanks i think i got it now... just confused about the algebra

 

[fredrick08]

all solved, thanks very much

 

[larrykuhl]

I'm still at a loss as to how n = 1.272.

so getting to:
1/n = sin(arctan(n))

then squaring and adding cos^2 gets:

1/n^2 - cos^2(arctan(n)) = 1

but any way I rearrange the above formula to get 1/cos^2 for sec^2 ends up resulting in n=1?