Exact Values for the Sine and Cosine of Multiples of 18 degrees: A Geometric Approach

Questions involving the exact values of the trigonometric functions of 180, 360, 540 and 720 have been popular in compilations of challenging mathematical problems, and in the problems sections of mathematics journals. (See, for example, problems 189 and 339 in [1] and the sections entitled Trigonometry: numerical evaluation, Trigonometry: numerical identities and Trigonometry: theory of equations in [2].) Invariably, the determination of these values make little or no use of geometry; rather, the derivations rely heavily upon trigonometric identities, such as the double and triple angle formulas or the product-to-sum formulas. In this note, we will use a purely geometric approach to derive the values of the sine and cosine of multiples of 18?. Therefore, this capsule can be used as enrichment or investigative material for students in trigonometry or precalculus courses. Since 0? and 900 are already standard trigonometric angles, we consider only angles of measure 18?, 36?, 54? and 720. Using Figure 1, we first show that the sine and cosine of 18?, 36?, 540 and 72? can all be expressed as functions of x. Toward this end, draw the altitude CD from vertex C to the side AB (Figure(b)), and l n et BD = y. For the right triangle BCD, we find (by the Pythagorean theorem) that CD = x2 y2, and for the right triangle ADC, we see that CD = 1 (1 y)2 = 2y y2 Equating the expressions for CD leads to y = x2. Therefore