11triangle is equilateral if all angles r 60°
tan A+tan B + tan C = 3√3
so wat was the logic of this question
do we have to prove something???
Prove that a triangle ABC is equlateral if and only if
tan A+tan B + tan C = 3√3
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12 Answers
1111Tan 60 = √3
Equilateral therefore A=B=C =60
√3+√3+√3
= 3√3
Hence proved
1tan A+ tan B+tan C=√3+√3+√3
if the triangle is equilateral...then angles r 60.........
therfore..........
3tan60=3√3
tan60=√3..............
62ALl of you have only proved that equilateral triangel implies this sum is 3 root 3
this was an 8 mark question..
there has to be more to it :P
62you have to prove that if the given expression holds then the triangle is equilateral.
1do we have to take the expression........
formula..........of the form tanx+tany or something bhaiyya....[56]..
1in a triangle
tanA+tanB+tanC=tanAtanBtanC
also tanA+tanB+tanC/3≥tanAtanBtanC (am≥gm)
equality holds gud only if tanA=tanB=tanC
3√3/3≥3√3√3
or √3≥√3
since equality is true tanA=tanB=tanC and triangle is equilateral
1amit first u only said
"equality holds gud only if tanA=tanB=tanC
"
and then u proved
"tanA=tanB=tanC" [7][7]
24no one is trying it correctly so i decided to solve a parallel problem.......
what if we had to prove that triangle is equilarteral using cosA+cosB+cosC=3/2
here is how to do it................
try with tan in a similar way.......u can do it.....[188]
1Let ABC be such that tanA+tanB+tanC=3√3
We should prove that ABC is equilateral
(tanA+tanB+tanC)/3=√3
For a triangle,tanA+tanB+tanC=tanAtanBtanC
tanAtanBtanC=3√3
3√tanAtanBtanC=√3
Apply AM GM for tanA+tanB+tanC
==> (tanA+tanB+tanC)/3 ≥3√tanAtanBtanC
But both sides are equal,
==> tanA=tanB=tanC
==>A=B=C