Math 2.0

Special functions Definition

Sine and cosine

Builds on Power series

Definition

The sine and cosine functions sin⁡,cos⁡ ⁣:ℝ→ℝ\sin, \cos \colon \mathbb{R} \to \mathbb{R} are defined by the power series sin⁡x=∑n=0∞(−1)nx2n+1(2n+1)!,cos⁡x=∑n=0∞(−1)nx2n(2n)!.\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}, \qquad \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}.

Remarks

Both series converge absolutely for every real xx. Sine is odd and cosine is even. Also sin⁡0=0\sin 0 = 0 and cos⁡0=1\cos 0 = 1. Geometrically, (cos⁡x,sin⁡x)(\cos x, \sin x) is the point of the unit circle at arc length xx from (1,0)(1, 0), counterclockwise. Mathlib defines Real.sin and Real.cos from the complex exponential, with cos⁡x+isin⁡x=eix\cos x + i \sin x = e^{i x}.

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Where this leads

37 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sine_and_cosine, Series and polynomials
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Tag0WB57D9488
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Kinddefinition
WritingW0 stub
MSC33B10, 26A09
DefinesSine and cosine
UsesPower series

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