2: Geometry, Probability, and Cardinality

Suppose a right triangle has edges A,B,C with corresponding lengths a, b, c. Suppose C is the hypotenuse. Let θ be the angle of the triangle formed by the edges A and C. From the definitions of trigonometric functions that we used in high school, cos θ = a/c and sin θ = b/c. Also, from an identity that we learned in high school, (cos θ) + (sin θ) = 1, so (a/c) + (b/c) = 1, i.e. a + b = c. However, if a + b = c, then we can reverse this reasoning to conclude that (cos θ) + (sin θ) = 1. So, it seems that the Pythagorean Theorem is more or less equivalent to the identity (cos θ) + (sin θ) = 1. To avoid circular reasoning, we would like to give a proof of the Pythagorean Theorem from first principles. In some sense the first proof will be roundabout, since a simpler proof could be given. However, the concepts that we introduce are actually extremely important, though this may not be clear right now. Since the rigorous treatment of limits is not a prerequisite of this course, we will not treat limits in a rigorous fashion. Try to find where these details are avoided. Let x ∈ R. We define sinx and cos x by the following formulas.