The diagram shows the path of a light ray through a triangular prism. Which of the following correctly relates the apex angle of the prism 𝐴 to the angles of incidence and the angles of refraction of the incident and emergent rays shown? (A) 𝜃 one plus 𝜃 two equals 𝐴. (B) Φ one plus Φ two equals 𝐴. (C) Φ one plus 𝜃 two equals 𝐴. (D) 𝜃 one plus Φ two equals 𝐴. And (E) 180 degrees minus the quantity 𝜃 one plus Φ two equals 𝐴.
In our diagram, we see our incident ray of light entering the triangular prism and then ultimately leaving it. The initial angle of incidence of this ray is Φ one, the initial angle of refraction is 𝜃 one, and then the second angle of incidence is Φ two and the second angle of refraction is 𝜃 two. We want to develop an expression for the apex angle 𝐴 in terms of these values. We’re going to be able to do this using a geometric approach.
Starting off, let’s clear some space on screen, and let’s note that the apex angle 𝐴 is part of this quadrilateral in orange. If we think about the corners of this four-sided shape, we know that this corner here is a right angle. That’s because it’s defined by a line normal or perpendicular to the face of the prism on this side. And for the same reason, this angle here in the quadrilateral is also 90 degrees.
At this point, let’s recall that for any four-sided shape, if we add together the angles of the four corners of that shape, we always get a total of 360 degrees. We can apply this fact to our quadrilateral in orange. The angle at the top of this shape is 𝐴. Then, there’s one 90-degree angle and a second 90-degree angle. And lastly, there’s this angle at the bottom of the shape. We’ll leave that label blank for now. Our rule for four-sided objects says these four angles added together must equal 360 degrees.
Notice on the left we have 90 degrees plus 90 degrees; that’s 180 degrees. And if we subtract this angle from both sides of the equation, then 180 degrees minus 180 degrees on the left goes to zero. And we have an expression that says that 𝐴 plus our unknown angle is equal to 180 degrees. If we now subtract the angle 𝐴 from both sides, canceling this angle on the left, we see that the unknown angle in our quadrilateral is 180 degrees minus 𝐴.
Knowing this, let’s now draw a zoomed-in version of this triangle that we’ve highlighted in pink. The angle at the top left of this triangle is 𝜃 one. The angle at the top right is Φ two. And we now know that the angle at the bottom is 180 degrees minus 𝐴. Similarly to what we found for our four-sided shape, if we consider a three-sided shape in general, a triangle, the sum of these three interior angles always equals 180 degrees. This means we can write that 𝜃 one plus Φ two plus 180 degrees minus 𝐴 equals 180 degrees. If we then subtract 180 degrees from both sides of this equation, that angle measure cancels out entirely.
What’s left is an expression that reads 𝜃 one plus Φ two minus 𝐴 equals zero. And if we then add 𝐴 to both sides of the equation so that 𝐴 cancels on the left, we find that 𝜃 one plus Φ two equals 𝐴. Reviewing our answer options, this matches option (D). This is how the apex angle 𝐴 can be expressed in terms of the angles of incidence and angles of refraction of this scenario. The initial angle of refraction, 𝜃 one, plus the second angle of incidence, Φ two, equals the apex angle 𝐴.