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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: 2020-09-23 15:18:00 | by: bookofproofs | references: [581]