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trigonometric formulas from series∗ pahio† 2013-03-22 2:41:04 One may define the sine and the cosine functions for real (and complex) arguments using the power series sin x = x − x3 x5 + − +..., 3! 5! (1) cos x = 1 − x4 x2 + − +..., 2! 4! (2) and using only the properties of power series, easily derive most of the goniometric formulas, without any geometry. For example, one gets instantly from (1) and (2) the values sin 0 = 0, cos 0 = 1 and the parity relations sin(−x) = − sin x, cos(−x) = cos x. Using the Cauchy multiplication rule for series one can obtain the addition formulas ( sin(x+y) = sin x cos y + cos x sin y, (3) cos(x+y) = cos x cos y − sin x sin y. These produce straightforward many other important formulae, e.g. sin 2x = 2 sin x cos x, cos 2x = cos2 x − sin2 x (y =: x) (4) and cos2 x + sin2 x = 1 (y =: −x). (5) ∗ hTrigonometricFormulasFromSeriesi created: h2013-03-2i by: hpahioi version: h41654i Privacy setting: h1i hDerivationi h26A09i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 π = 0, as well as the formulae expressing the periodicity of 2 sine and cosine, cannot be directly obtained from the series (1) and (2) — in fact, one must define the number π by using the function properties of the cosine series and its derivative series. The equation cos x = 0 The value cos has on the interval (0, 2) exactly one root. Actually, as sum of a power series, 4 2 cos x is continuous, cos 0 = 1 > 0 and cos 2 < 1 − 22! + 24! < 0 (see Leibniz’ estimate for alternating series), whence there is at least one root. If there were more than one root, then the derivative − sin x = −x + x3 x2 − + . . . = −x(1 − + − . . .) 3! 3! would have at least one zero on the interval; this is impossible, since by Leibniz the series in the parentheses does not change its sign on the interval: 1− x2 22 + −... > 1 − > 0 3! 3! Accordingly, we can define the number pi to be the least positive solution of the equation cos x = 0, multiplied by 2. Thus we have 0 < π < 4 and cos π2 = 0. Furthermore, by (5), sin π = 1, 2 and by (4), sin π = 0, cos π = −1, sin 2π = 0, cos 2π = 1. Consequently, the addition formulas (3) yield the periodicities sin(x+2π) = sin x, cos(x+2π) = cos x. 2