What is the significance of commas within equation solution output?

Questions
,
1

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]]))]
2

Multiple solutions?

Solve[x^2 == 1, x]

{{x -> -1}, {x -> 1}}
3

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.

4

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.

5

Called with 2 arguments ArcTan[y,x] is equivalent to ArcTan[y/x} but takes into acount the quandrant of the point {x,y}.
Best,
David

6

I’ve got this figured out now, thanks for your feedback.