Abstract
But how is it that the sines of these angles can be obtained via such a simple arithmetic sequence? The angles themselves occur in a pattern to be sure, but not such a simple one. We will show that θ (see the figure) is one of the special angles, 0, π/6, π/4, π/3, or π/2, precisely when the ratio |AD| |AC| is 0, 1/4, 1/2, 3/4, or 1, respectively. To put it another way, if we think of the point D as moving in a straight line from A to C then θ successively reaches each of the special angles π/6, π/4, and π/3 precisely when D is 1/4, 1/2, 3/4, or all of the way to C . (Clearly θ = {0, π/2} when D = {A,C}.) For clarity and without loss of generality we assume that |AC| = 1. Observe that triangles ABD and BCD are similar to each other since both are similar to triangle ABC. Thus by proportionality of similar triangles we have |BD| |AD| = 1−|AD| |BD| or
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