Mathematica 9 is now available
Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2007
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2007

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: showing an expression equal to 0

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73719] Re: showing an expression equal to 0
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
  • Date: Mon, 26 Feb 2007 06:14:52 -0500 (EST)
  • References: <errm70$8bp$1@smc.vnet.net>

"dimitris" <dimmechan at yahoo.com> wrote:
> Consider
>
> In[87]:=
> expr = ArcTan[y/Sqrt[R^2 - y^2]] - ArcSin[y/R];
>
> The expression for R>0 and -R<y<R is equal to zero
>
> In[72]:=
> (Plot[Chop[Evaluate[expr /. R -> #1]], {y, -#1, #1}, Axes -> False,
> Frame -> True] & ) /@ Range[5]
>
> Based on Andrzej's resposnse on a recenet message I am able to show
> this equality for given R.
>
> E.g. for R=4 the following demonstrates that expr is equal to zero
>
> In[99]:=
> Simplify[expr /. R -> 4 /. y -> 2]
>
> Out[99]=
> 0
>
> In[97]:=
> D[expr /. R -> 4, y]
> Factor[%]
>
> Out[97]=
> -(1/(4*Sqrt[1 - y^2/16])) + (y^2/(16 - y^2)^(3/2) + 1/Sqrt[16 - y^2])/
> (1 + y^2/(16 - y^2))
> Out[98]=
> 0
>
> However not specifying R (but assuming R>0&&-R<y<R) I am not able to
> show that expr is equal to zero.
>
> Any ideas?

In[4]:= Simplify[R Sin[ArcTan[y/Sqrt[R^2 - y^2]]] == y, -R<y<R]

Out[4]= True

Since we also know that ArcTan and ArcSin have the same range, we can
conclude that ArcTan[y/Sqrt[R^2 - y^2]] == ArcSin[y/R]  for  -R < y < R.
A similar method is

In[5]:=
Simplify[TrigExpand[Sin[ArcTan[y/Sqrt[R^2 - y^2]] -  ArcSin[y/R]]], -R<y<R]

Out[5]= 0

David


  • Prev by Date: Re: Tr[list] vs Plus@@list
  • Next by Date: Re: Symbol name extraction
  • Previous by thread: Re: showing an expression equal to 0
  • Next by thread: Re: showing an expression equal to 0