PlanetMath: sinusoid

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sinusoid

(Definition)

A sinusoid is a curve of the form

$\displaystyle \mathbbmss{R}$	$\displaystyle \to$	$\displaystyle \mathbbmss{R}^2$
$\displaystyle t$	$\displaystyle \mapsto$	$\displaystyle (t, \sin{kt}),$

where

is a parameter determining the oscillation.

The basic sinusoid, the curve

$\displaystyle y = \sin{x}$

in the

-plane, oscillates periodically with the period of sine, $2\pi$ , as

increases.

$\includegraphics{sinusoid}$

On the interval $[0, \frac{\pi}{2}]$ , the curve is ascending because the derivative of sine, $\cos{x}$ , is positive for acute angles .
Consequently, on the interval $[\frac{\pi}{2}, \pi]$ ,the supplement formula $\sin{(\pi-x)} = \sin{x}$ tells that the sinusoid is descending.
Thus we get on the whole interval $[0, \pi]$ a cap-formed ( $\smallfrown$ ) arc.
Because sine is an odd function, we have on the interval $[-\pi, 0]$ the mirror image of the cap, a cup-formed ( $\smallsmile$ ) arc.
All in all, on the period interval $[-\pi, \pi]$ the sinusoid consists of the consecutive cup and cap, together a lying-S formed ( $\backsim$ ) arc.
The same is repeated on each other period interval $[(2n\!-\!1)\pi, (2n\!+\!1)\pi]$ where $n\in\mathbb{Z}$ .

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"sinusoid" is owned by matte. [ full author list (3) ]

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Cross-references: consecutive, period, arc, supplement formula, acute angles, positive, interval, parameter, curve
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This is version 8 of sinusoid, born on 2005-05-22, modified 2008-11-14.
Object id is 7101, canonical name is Sinusoid.
Accessed 2014 times total.

Classification:

53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

Pending Errata and Addenda

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