Question

The answer to problem 2C-10 is in terms of the sines of the angles but the ratios that result from differentiation are cosines aren't they?

Answer 1

I will tell you I got the same thing and its been driving me crazy. It uses the segment adjacent to the angle and the hypotenuse and that is cos. Its nice to see that someone else got that too. I just wish that meant we were both right :).

Answer 2

You're correct, the exercise is faulty. In fact it's faulty in two ways. The one that relates to the question posed here is that Snell's law relates to the angle of incidence, which is the angle made by the line along which the runner/swimmer travels with a line VERTICAL to the shoreline, not the angle between the line of travel and the shoreline. Because cos x = sin (pi/2 -x), this mislabeling of the angles translates Snell's law into a statement about the cosines of the angles labeled . . . but these are not the angles of incidence.

The other mistake here is that Snell's law has the sines of the angles of incidence (and therefore the cosines of the mislabeled angles in this exercise) DIRECTLY proportional to the velocities, not inversely proportional (and that is what the calculations show). In optics, the sines are inversely proportional to the refractive indexes of the mediums, but proportional to the velocities.