sinusoid

A sinusoid is a curve of the form

$ℝ$$\to$${ℝ}^2$

$t$$\mapsto$$(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 ($⌢$) arc.

• 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 cup-formed ($⌣$) 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  $n\in \mathbb{Z}$.