Somewhat More than Governors Need to Know about Trigonometry

Abstract

SummaryWe observed that we can construct a trig table for each natural number d by including precisely those rational angles θ such that cos θ and sin θ algebraic of degree at most d. For d = 1, the table only included multiples of 90°. For d = 2, we got the standard trig table consisting of multiples of 30° and 45°. For d = 4, we found a larger table, and we exhibited a portion of it in Table 2. Unfortunately, for d ≥ 6, it is impossible to write the cosine of some angles without using complex numbers. Further investigation revealed that, without using complex numbers, we can only write down cos θ and sin θ if φ(n)is a power of 2, where n is the denominator of θ. The proof of this last fact was different from the similar-sounding result about constructibility of regular n-gons with straightedge and compass.