# sinusoid

A sinusoid is a curve of the form

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

where $k>0$ is a parameter determining the oscillation.

The basic sinusoid, the curve

 $y=\sin{x}$

in the $xy$-plane, oscillates periodically with the period of sine (http://planetmath.org/ComplexSineAndCosine), $2\pi$, as $x$ increases.

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

• 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 (http://planetmath.org/ComplexSineAndCosine), we have on the interval  $[-\pi,\,0]$  the 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  $n\in\mathbb{Z}$.

Title sinusoid Sinusoid 2015-02-04 11:23:26 2015-02-04 11:23:26 matte (1858) pahio (2872) 12 matte (2872) Definition msc 53A04 Trigonometry DefinitionsInTrigonometry