sinusoidSinusoid2005-05-22 12:45:512008-11-14 19:30:29Definitionfamous curves% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
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\newtheorem{cor}{Corollary}A \emph{sinusoid} is a curve of the form
\begin{eqnarray*}
\R&\to& \R^2 \\
t&\mapsto & (t,\,\sin{kt}),
\end{eqnarray*}
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 \PMlinkname{period of sine}{ComplexSineAndCosine}, $2\pi$, as $x$ increases.
\begin{center}
\includegraphics{sinusoid}
\end{center}
\begin{itemize}
\item On the interval \,$[0,\,\frac{\pi}{2}]$,\, the curve is ascending because the \PMlinkname{derivative of sine}{ComplexSineAndCosine}, $\cos{x}$, is positive for acute angles $x$.
\item Consequently, on the interval \,$[\frac{\pi}{2},\,\pi]$,\,the supplement formula \,$\sin{(\pi-x)} = \sin{x}$\, tells that the sinusoid is descending.
\item Thus we get on the whole interval \,$[0,\,\pi]$\, a cap-formed ($\smallfrown$) arc.
\item Because \PMlinkname{sine is an odd function}{ComplexSineAndCosine}, we have on the interval \,$[-\pi,\,0]$\, the \PMlinkescapetext{mirror image} of the cap, a cup-formed ($\smallsmile$) arc.
\item 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.
\item The same is repeated on each other period interval \,$[(2n\!-\!1)\pi,\,(2n\!+\!1)\pi]$\, where \,$n\in\mathbb{Z}$.
\end{itemize}