Math Is Fun Forum

4.   A triangle ABC
The angles achieve the following relationship :Sin(2A)+sin(2B)=2Sin(C)
what kind of this triangle?

hi bob ;

Excellent! q2 is correct

Q1. the method that i know

sin(A)=sin(B).cos(C)

so: sin(B+C)=sin(B).cos(C)

sin(B).cos(C) + cos(B). sin(C)= sin(B).cos(C)

==>  cos(B). sin(C)=0

Either :  sin(C)=0    =>   C=0    ( Rejected,because (C)is an angle in the triangle)   

or :     cos(B)=0    =>   B= π/2 rad   

thus,it is a right triangle .

a simple method  :

the gray area is   3( r^2 - 1/4 π r^2)

the pink area is 1/4π r^2 -(π-3) r^2

= r^2(1/4π-π+3)

=r^2(-3/4π+3)

=3 r^2(1-1/4π)

=3(r^2-1/4π r^2)=the gray area

thank you very much Mr Bob and Mr Bobbym

 
what should i do to provide an image ?

sorry but  the image doesn't appear

Thank you bob