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Trig Function Topics

Trig Functions: Sine and Cosine Definition
Definition: An algebraic approach From defining a few general properties of the sine and cosine functions, we can alegbraically derive the sine and cosine functions themselves. First, define the sine and cosine functions to have these properties:
sin 0 = 0 = a_{0} a_{0} = 0 By differentiating the power series and equating it with the cosine by the original properties, cos 0 = 1 = (1)a_{1} a_{1} = 1 Continuing, sin 0 = 0 = (2)a_{2} a_{2} = 0 sin x = cos x = (0 + 0 + 0 + (3)(2)(1)a_{3}x^{0} + (4)(3)(2)a_{4}x^{1} + ...) cos 0 = 1 = (3)(2)(1)a_{3} a_{3} = 1/3! cos x = sin x = 0 + 0 + 0 + 0 + (4)(3)(2)(1)a_{4}x^{0} + ... sin 0 = 0 = (4)(3)(2)(1)a_{4} a_{4} = 0 Continue on and you get values of all a_{n} if n is even then a_{n} = 0 if n is odd then a_{n} = 1/n! alternating positive and negitive. or stated as: a_{2n} = 0 a_{2n+1} = (1)^{n}/(2n+1)! Pluging these values into the equation for sine: sin x = x^{1}/1!  x^{3}/3! + x^{5}/5!  x^{7}/7! + x^{9}/9!  ... + (1)^{n}x^{2n+1}/(2n+1)! + ... = (1)^{n}x^{2n+1} / (2n+1)! and for cosine: cos x = 1  x^{2}/2! + x^{4}/4!  x^{6}/6! + x^{8}/8!  ... + (1)^{n}x^{2n}/(2n)! + ... = (1)^{n}x^{2n} / (2n)! These series converge for all x ÎÂ. Source: Jeff Yates. 