Question

If x and y are the other angles of a right angled triangle, show that sin2x = sin2y?

Answer 1

`"see explanation"`

Explanation:

`"since the triangle is right angled then"`

`x+y=90rArry=90-x`

`rArrsiny=sin(90-x)=cosx`

`rArrcosy=cos(90-x)=sinx`

`•color(white)(x)sin2x=2sinxcosx`

`rArrsin2y=2sinycosy=2cosxsinx`

`rArrsin2x=sin2y`

Answer 2

`sin(2x)=sin(2y)=(2* (AB).(BC))/(AC)^2`

Explanation:

Let `DeltaABC` be a right triangle `/_B=90^0 , /_C=x^0`

and `/_A=y^0; BC` is base , `AB` is perpenficular and

`AC` is hyotenuse. `Sin x=(AB)/(AC), cos x=(BC)/(AC)` ,

`Sin y=(BC)/(AC), cos y=(AB)/(AC) ; sin(2x)=2sinxcosx`

`=2*(AB)/(AC).(BC)/(AC)= (2* (AB).(BC))/(AC)^2`

`sin(2y)=2sinycosy=2*(BC)/(AC).(AB)/(AC)= (2* (AB).(BC))/(AC)^2`

`:. sin(2x)=sin(2y)=(2* (AB).(BC))/(AC)^2` (Proved) [Ans]