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

Corollaries: 1

  1. Proposition: Oddness of the Sine of a Real Variable
  2. Proposition: Derivative of Sine
  3. Proposition: Integral of Sine

Corollaries: 1 2 3 4 5
Definitions: 6
Proofs: 7 8 9
Propositions: 10 11 12 13 14 15 16 17 18 19
Theorems: 20

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983