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What is the significance of commas within equation solution output?

Posted 11 years ago
I solved the following equations with Mathematica and used the Simplify function to condense
the solutions.  The simplified solution for the variable "Alpha" is included below.  What I don't understand
is the significance of the comma in the solution (located just before the break in the code).  Mathematica is
fairly new to me, and I am unfamiliar with the meaning of a comma in this context.  I would greatly appreciate
any feedback related to this matter.  
 Solve[EG + (CD + DE) Cos[\[Delta]] + BC Cos[\[Alpha]] +
 BG Cos[\[Gamma]] ==  0 &&
 (CD + DE) Sin[\[Delta]] + BC Sin[\[Alpha]] + BG Sin[\[Gamma]] == 0, {\[Alpha], \[Gamma]}]  
 ArcTan[-(BC^3 (EG + (CD + DE) Cos[\[Delta]]) + BC (EG + (CD + DE) Cos[\[Delta]]) (-BG^2 +
 (CD + DE)^2 + EG^2 +  2 (CD + DE) EG Cos[\[Delta]]) + \(-BC^2 (CD + DE)^2 (BC^4 +
 BG^4 + (CD + DE)^4 + 4 (CD + DE)^2 EG^2 +  EG^4 - 2 BG^2 ((CD + DE)^2 + EG^2) -
 2 BC^2 (BG^2 + (CD + DE)^2 + EG^2) +  2 (CD + DE) EG (2 (-BC^2 - BG^2 + (CD + DE)^2
 + EG^2) Cos[\[Delta]] + (CD + DE) EG Cos[2 \[Delta]])) Sin[\[Delta]]^2))/(BC^2 ((CD + DE)^2
+  EG^2 + 2 (CD + DE) EG Cos[\[Delta]])), 

(-BC (CD + DE)^2 (BC^2 -BG^2 + (CD + DE)^2 +
EG^2 + 2 (CD + DE) EG Cos[\[Delta]]) Sin[\[Delta]] + (EG + (CD + DE) Cos[\[Delta]]) Csc[\[Delta]] \
(-BC^2 (CD + DE)^2 (BC^4 + BG^4 + (CD + DE)^4 + 4 (CD + DE)^2 EG^2 + EG^4 - 2 BG^2
((CD + DE)^2 + EG^2) - 2 BC^2 (BG^2 + (CD + DE)^2 + EG^2) + 2 (CD + DE) EG (2 (-BC^2 -
BG^2 + (CD + DE)^2 + EG^2) Cos[\[Delta]] + (CD + DE) EG Cos[2 \[Delta]])) Sin[\[Delta]]^2))
/(BC^2 (CD + DE) ((CD + DE)^2 + EG^2 + 2 (CD + DE) EG Cos[\[Delta]]))]
POSTED BY: Ryan Gardner
5 Replies
Posted 11 years ago
I've got this figured out now, thanks for your feedback.
POSTED BY: Ryan Gardner
Posted 11 years ago
Called with 2 arguments ArcTan[y,x] is equivalent to ArcTan[y/x} but takes into acount the quandrant of the point {x,y}.
POSTED BY: David Keith
Posted 11 years ago
Here is a simplified version of your result which I think explains the two kinds of commas which appear.
In[1]:= Solve[EG+(CD+DE)Cos[\[Delta]]+BC Cos[\[Alpha]]+BG Cos[\[Gamma]]==0 &&
                 (CD+DE)Sin[\[Delta]]+BC Sin[\[Alpha]]+BG Sin[\[Gamma]]==0, {\[Alpha],\[Gamma]}]

Out[1]= {{\[Alpha]->ConditionalExpression[ArcTan[numerator1,denominator1]+2 \[Pi] C[1],C[1] \[Element] Integers],
          \[Gamma]->ConditionalExpression[ArcTan[numerator2,denominator2]+2 \[Pi] C[2],C[2] \[Element] Integers]},
         {\[Alpha]->ConditionalExpression[ArcTan[numerator3,denominator3]+2 \[Pi] C[1],C[1] \[Element] Integers],
          \[Gamma]->ConditionalExpression[ArcTan[numerator4,denominator4]+2 \[Pi] C[2],C[2] \[Element] Integers]}}
There are two different forms of ArcTan in Mathematica, ArcTan and ArcTan[x,y] where the second form can express exactly which quadrant the point is in.
That accounts for the four commas inside the four ArcTan[] expressions.

ConditionalExpression[Value, BooleanConditionWhichMustBeTrueForThisToBeTheValue] accounts for the four commas outside the ArcTan.

I think that accounts for all of the commas.
POSTED BY: Bill Simpson
Posted 11 years ago
The solution was preceeded by \ -> ConditionalExpression[. Given this I would expect the output after
the comma to be a conditional statement, but it does not appear to be a logical statement of any kind. The
output before  the comma does not seem to be equivalent to the output after the comma, and so I doubt that
the comma is used to distinguish between two different solutions. In any case, the entire expression (comma
included) yields correct results when numeric values are plugged into it.
POSTED BY: Ryan Gardner
Multiple solutions?
Solve[x^2 == 1, x]
{{x -> -1}, {x -> 1}}
POSTED BY: Frank Kampas
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