PlanetMath: phase parametrization in $\cos(x+p)=\cos x$

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

phase parametrization in $\cos(x+p)=\cos x$

(Definition)

We obtain a single equation to parametrization of phase's angle for the periodic cosine function when the typical condition $\cos(x+p)=cos x$ is satisfied. Without losing generality, we will restrict the discussion to the case $0\leq x \leq 2\pi$ . We starting setting $\cos x=t$ , so that $-1\leq t\leq 1$ is the interval of definition for the real parameter . Next we expand out

$\displaystyle \cos(x+p)=\cos x\cos p-\sin x\sin p,$

where we introduce

and apply the assumed condition to get

$\displaystyle t\cos p-\sqrt{1-t^2}\sin p=t.$

This leads to a quadratic equation in $\cos p$ on terms of parameter

, i.e.

$\displaystyle \cos^2p(t)-2t^2\cos p(t)+(2t^2-1)=0.$

Solving,

$\displaystyle \cos p(t)=t^2\pm (t^2-1).$

One root has to do with the trivial $\cos(x+2\pi)=\cos x$ , but we are interested on the nontrivial one

$\displaystyle \cos p(t)=2t^2-1, \qquad -1<t<1.$

(1)

Locus of (1) is the below shown parabola.

unit=2cm
$\begin{pspicture}(-2.5,-2)(2.5,5) \psaxes[Dx=1,Dy=1]{->}(0,0)(-2.2,-1.5)(2.3,2.5... ...^2-1$} \psdot(-0.707,0) \psdot(0.707,0) \psdot(-1,1) \psdot(1,1) \end{pspicture}$

Inverse function of (1) is the wanted parametrization , with codomain $[0,\pi]$ and locus the “arabian dome” below shown.¹

unit=2cm
$\begin{pspicture}(-2.5,-2)(2.5,4) \psaxes[Dx=1,Dy=1]{->}(0,0)(-2.2,-1.5)(2.3,3.7... ...arccos 3.1416 mul 180 div} \rput(1.8,2.2){$p = \arccos(2t^2-1)$} \end{pspicture}$

Footnotes

...¹: Graphics by pahio, with permission.

"phase parametrization in $\cos(x+p)=\cos x$ " is owned by perucho.

(view preamble)

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: codomain, inverse function, parabola, locus, root, terms, quadratic equation, expand, parameter, real, interval, function, cosine, periodic, angle, equation

This is version 1 of phase parametrization in $\cos(x+p)=\cos x$ , born on 2008-03-22.
Object id is 10434, canonical name is PhaseParametrizationInCosxpcosX.
Accessed 127 times total.

Classification:

AMS MSC:	33B10 (Special functions :: Elementary classical functions :: Exponential and trigonometric functions)
	42-00 (Fourier analysis :: General reference works )
	51-00 (Geometry :: General reference works )
	43-00 (Abstract harmonic analysis :: General reference works )

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact