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Definition: Cosine of a Real Variable

Let \(x\in\mathbb R\) be any real number and let \(z\) be the complex number obtained from \(x\) by multiplying it with the imaginary unit, i.e. \(z:=ix\).

The cosine of \(x\) is a function \(f:\mathbb R\mapsto\mathbb R\), which is defined as the real part of the complex exponential function

\[\cos(x):=\Re(\exp(ix)).\]

Geometrically, the cosine is a projection of the complex number \(\exp(ix)\), which is on the unit circle, to the real axis. The behavior of the cosine function can be studied in the following interactive figure (with a draggable value of \(x\)):

Cosine graph of $\cos(x)$

Projection of $\exp(ix)$ happening in the complex plane

| | | | | created: 2016-02-28 18:24:40 | modified: 2019-09-14 09:26:40 | by: bookofproofs | references: [581]

1.Proposition: Eveness of the Cosine of a Real Variable

2.Proposition: Zero of Cosine

3.Proposition: Derivative of Cosine

4.Proposition: Integral of Cosine

5.Corollary: Representing Real Cosine by Complex Exponential Function

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983