A sinusoid is a curve of the form

t (t,sinkt),

where k>0 is a parameterMathworldPlanetmath determining the oscillation.

The basic sinusoid, the curve


in the xy-plane, oscillates periodically with the period of sine (, 2π, as x increases.

  • On the intervalMathworldPlanetmathPlanetmath[0,π2],  the curve is ascending because the derivative of sine (, cosx, is positive for acute anglesMathworldPlanetmathPlanetmath x.

  • Consequently, on the interval  [π2,π], the supplement formulasin(π-x)=sinx  tells that the sinusoid is descending.

  • Thus we get on the whole interval  [0,π]  a cap-formed () arc.

  • Because sine is an odd functionMathworldPlanetmath (, we have on the interval  [-π, 0]  the of the cap, a cup-formed () arc.

  • All in all, on the period interval  [-π,π]  the sinusoid consists of the consecutive cup and cap, together a lying-S formed () arc.

  • The same is repeated on each other period interval  [(2n-1)π,(2n+1)π]  where  n.

Title sinusoid
Canonical name Sinusoid
Date of creation 2015-02-04 11:23:26
Last modified on 2015-02-04 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 TrigonometryMathworldPlanetmath
Related topic DefinitionsInTrigonometry