In this case, we're going to have refraction across two boundaries.

Here is a block of glass.

Index of refraction is 1.5.

We have our first boundary, we have our second boundary.

Here's the incoming ray.

It goes through the glass and then out the glass on the other side.

Notice that I used the numerics that the angle of incidence is theta sub 1, The angle of refraction across the first boundary is theta sub 2, then the angle of incidence on the second boundary is theta sub 3, and the angle of refraction across the second boundary is theta sub 4.

In line with that, the index of refraction on this side of the first boundary is n sub 1, the index of refraction on the other side of that boundary is n sub 2, the index of refraction on this side of the second boundary is n sub 3, which by the way of course is the same as n sub 2, as you're still inside the glass.

And then the index of refraction on the second side of the second boundary is n sub 4, which is back to what we started with, with the air.

Notice that when we have the first boundary, we're traveling from an index of refraction, which is low, to an index of refraction, which is high, which means that the beam bends or refracts towards the normal.

Here we're traveling from an index of refraction, which is high, to an index of refraction, which is slow, which means that the beam will refract or bend away from the normal.

And so the question is, what will theta sub 4 be equal to?

To figure that out, we have to start with theta sub 1.

Now theta sub 1 was not given, you're only given this angle right here.

And to help you out, we're going to continue drawing this line horizontally this way.

Now notice that this line right here is parallel to this line right there.

And this angle here and this angle here are what we call alternate interior angles, which means that if this angle is 45 degrees, then this angle must be 45 degrees as well.

And since this here is a right angle by definition, because this is the normal to this side right here, if this is 45 degrees, then this must be 45 degrees as well, which means theta sub 1 is 45 degrees.

And now we can go ahead and try to figure out what theta sub 2 is equal to.

And for that, we use Snell's Law, n1 sine of theta 1 is equal to n2 sine of theta sub 2.

Since we're looking for theta sub 2, we're going to flip the equation around.

So n2 sine of theta 2 equals n1 sine of theta 1.

Divide both sides by n sub 2, so we get the sine of theta sub 2 is equal to n1 over n2 times the sine of theta 1.

And finally, theta sub 2, therefore, is equal to the arc sine or the inverse sine of n1 over n2 times the sine of theta sub 1.

And we plug in the numbers, it's the arc sine of n1 is equal to 1 and 2 is equal to 1.5.

And notice how important it is that you do all the notations correctly right from the start.

It makes it a lot easier to plug the numbers into the equation.

And of course, sine of theta 1 is the sine of 45 degrees.

And that is equal to, and my calculator is right here.

So 45, take the sine, divide by 1.5, and take the arc sine of that.

And we have 28.1 degrees, so 28.1 degrees for theta sub 2.

All right.

Now, if theta sub 2 is, and I'll write over here, is 28.1 degrees, how big is theta sub 3?

Hmm.

Well, again, we use the same principle we did over here.

Notice that this line right here is parallel to this line right there.

And then we have the line coming across as the beam of light, which means that theta sub 3 here is the same as this angle right there.

So this angle and this angle are equal to each other.

Hmm.

So what is this angle equal to?

Well, let's see here.

We know that this angle from there to there is 90 degrees.

So this is a 90 degree angle.

We also know that this here must be a 45 degree angle.

Because these are opposite angles to each other.

So this angle is 45 degrees, that means this angle is 45 degrees.

And that means that the sum of this angle plus theta sub 2 is also 45 degrees, which means if we take down the total, 45 degrees, and subtract from that theta sub 2, we get this angle right there.

And since these are alternate interior angles, that means theta sub 3 must also be 45 degrees minus theta sub 2.

So theta sub 3 is equal to 45 degrees minus theta sub 2.

Alright, and since theta sub 2 is 28.1 degrees, that means theta sub 3 is equal to 45 degrees minus 28.1 degrees.

And so 45 minus 28.1 equals theta sub 3 is equal to, let's do that again, 45 minus 28.1 equals 16.9 degrees.

Alright, so now we have theta sub 3, which should allow us to find theta sub 4 again using Snell's Law.

So we have n1 sine of, oops, not n1, we're now going from 3 to 4, so let's call this n3.

So n3 sine of theta 3 is equal to n4 sine of theta sub 4.

We're looking for theta sub 4, so we can flip the equation around.

Sine of n4 sine of theta sub 4 equals n3 sine of theta sub 3.

And so sine of theta sub 4 is equal to, when we divide both sides by n sub 4, we get n sub 3 divided by n sub 4 times the sine of theta sub 3.

And finally, we take the arc sine, and so we have theta sub 4 is equal to the arc sine or inverse sine of n3 over n4 times the sine of theta sub 3.

And plug in the numbers, that's equal to the arc sine of n3, which is 1.5, n4, which is 1, times the sine of theta sub 3, which we said was 16.9 degrees.

Alright, so we take the sine of that, we multiply it times 1.5, and we take the arc sine of that number, and we get 25.6 degrees.

So theta sub 4 equals 25.6 degrees, and that's our answer.

Alright, so you can see that the most difficult part of this problem is trying to find all the angles.

And of course, it's getting very busy in here, but let's quickly recap how we did that.

We're given the angle of 45 degrees here, so we assume that's also a 45 degree angle over here.

This is a 90 degree angle.

We draw the ray, the ray was coming in horizontally, so parallel to the bottom.

And so, if we look at this side, and we look at this line and this line, we can call these alternate interior angles, so they must be equal to each other.

That means that this angle must also be 45 degrees, because this is a 90 degree angle.

So we determine theta sub 1.

Then we calculated theta sub 2, which is the refracted angle by using Snell's Law.

Then we have to find out what theta sub 3 was equal to.

And to do that, we realize that this here is a 90 degree angle.

This was a 45 degree angle, so we know that this whole thing here was a 45 degree angle, which means that this angle here is 45 degrees minus theta sub 2.

And then if we draw this line here and this line there, we see that theta sub 3 and this angle are alternate interior angles, so they must be equal to each other right here.

And since we knew what theta sub 2 was, we subtract from 45 degrees to find theta sub 3.

And then we use that in our equation right here to find theta sub 4.

And that's how you do that problem.