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Definition: Sine 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 sine of $x$ is a function $f:\mathbb R\mapsto\mathbb R,$ which is defined as the imaginary part of the complex exponential function

$$\sin(x):=\Im(\exp(ix)).$$

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

Sine Graph of $x$

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

| | | | | created: 2016-02-28 18:38:29 | modified: 2020-09-23 15:18:50 | by: bookofproofs | references: [581]

1.Proposition: Oddness of the Sine of a Real Variable

2.Proposition: Derivative of Sine

3.Proposition: Integral of Sine

4.Corollary: Representing Real Sine 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