sinusoid
A sinusoid is a curve of the form
$\mathbb{R}$  $\to $  ${\mathbb{R}}^{2}$  
$t$  $\mapsto $  $(t,\mathrm{sin}kt),$ 
where $k>0$ is a parameter^{} determining the oscillation.
The basic sinusoid, the curve
$$y=\mathrm{sin}x$$ 
in the $xy$plane, oscillates periodically with the period of sine (http://planetmath.org/ComplexSineAndCosine), $2\pi $, as $x$ increases.

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On the interval^{} $[0,\frac{\pi}{2}]$, the curve is ascending because the derivative of sine (http://planetmath.org/ComplexSineAndCosine), $\mathrm{cos}x$, is positive for acute angles^{} $x$.

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Consequently, on the interval $[\frac{\pi}{2},\pi ]$, the supplement formula $\mathrm{sin}(\pi x)=\mathrm{sin}x$ tells that the sinusoid is descending.

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Thus we get on the whole interval $[0,\pi ]$ a capformed ($\u2322$) arc.

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Because sine is an odd function^{} (http://planetmath.org/ComplexSineAndCosine), we have on the interval $[\pi ,\mathrm{\hspace{0.17em}0}]$ the of the cap, a cupformed ($\u2323$) arc.

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All in all, on the period interval $[\pi ,\pi ]$ the sinusoid consists of the consecutive cup and cap, together a lyingS formed ($\backsim $) arc.

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The same is repeated on each other period interval $[(2n1)\pi ,(2n+1)\pi ]$ where $n\in \mathbb{Z}$.
Title  sinusoid 

Canonical name  Sinusoid 
Date of creation  20150204 11:23:26 
Last modified on  20150204 11:23:26 
Owner  matte (1858) 
Last modified by  pahio (2872) 
Numerical id  12 
Author  matte (2872) 
Entry type  Definition 
Classification  msc 53A04 
Related topic  Trigonometry^{} 
Related topic  DefinitionsInTrigonometry 