Question

Referring to Problem Set 4, Applications of differentiation. 2C-10. Isn't the ratio x/sqrt(100^2+x^2) cosine alpha??

Answer 1

Yes, this exercise is messed up in a number of ways, and as a result we calculate the cosine of the indicated angle, not the sine. I've notified people at the program of this error.

The main problem here is that they started with the wrong diagram. The angle labeled alpha isn't the angle of incidence, which is actually the angle between the path of the runner/swimmer and the "normal,", that is, a line vertical to the shoreline. The angle they labeled alpha is actually complementary to the angle of incidence, and because the cosine of an angle is equal to the sine of its complement, we end up proving something about the cosines of the angles they labeled, not the sines of the angles of incidence. Note also that the ratio of the cosines is the ratio of the speeds, not the reciprocal of the ratio of the speeds as the exercise states.

Answer 2

Of course (facepalm)! That is not the angle of incident. I redrew the diagram to show correct angle of incident which would indeed yield sine ratio. Thanks for noting reciprocal error as well.