k x2k+1 ? (2k + 1)! and sin(x) Key Observation: Suppose that f is a function for which f(n)(a) exists for each n and hence with Taylor series f(x)

sin2 + cos2 = 1 (1) 1 + cot2 = cosec2 (2) tan2 + 1 = sec2 (3) Note that (2) = (1)=sin2 and (3) = (1)=cos . Compound-angle formulae.

This is very useful information about the function sin(x) but it doesn’t tell the whole story. For example, it’s hard to tell from the formula that sin(x) is periodic.

Unit circle properties sin( x) = sin(x) tan( x) = tan(x) sin( + x) = sin(x) tan( + sin(2 x) = sin(x)

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The identity that will help us here is the Pythagorean identity since we can rewrite the last equation as (1 cos2 x) + 2 sin x + sin2 x = 0: But 1 cos2 x = sin2 x by the Pythagorean identity. So we rewrite this as …

Trigonometry Identities Quotient Identities:Reciprocal Identities: tan