In a triangle `A B C ,`if `acosA=bcosB ,`show that the triangle is either isosceles or right angled.

In a triangle `A B C ,`if `acosA=bcosB ,`show that the triangle is eit

1.	In a triangle `A B C ,`if `acosA=bcosB ,`show that the triangle is either isosceles or right angled.
Answer» By the sine rule, we have `a/("sin A")=b/("sin B")=c/("sin C")="k (sky)"` `rArr" "a=ksinAandb=ksinB` `:." "acosA=bcosB` `rArr" "ksinAcosA=ksinBcosB` `rArr" "1/2(sin2A)=1/2(sin2B)` `rArr" "sin2A=sin2B` `rArr" "sin2A-sin2B=0` `rArr" "2cos(A+B)sin(A-B)=0` `rArr" "cos(A+B)=0orsin(A-B)=0` rArr" "(A+B)=pi/2or(A-B)=0` `rArr" "/_C=pi/2orA=B.` Hence, `DeltaABC` is right-angled or isosceles.