# 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

Corollaries: 1

Corollaries: 1 2 3 4 5
Definitions: 6
Proofs: 7 8 9
Propositions: 10 11 12 13 14 15 16 17 18 19 20
Theorems: 21

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### References

#### Bibliography

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