Q. 7

In a right-angled triangle ABC, write the value of sin2 A + sin2 B + sin2 C.

Answer :

Given, triangle ABC is right angle.


So, let B = 90°


Then as per the property of angles in a triangle


A + B + C = 180°


As B = 90°


A + 90° + C = 180°


Then A + C = 180° - 90° = 90°


Now, consider sin 2A + sin 2B + sin 2C


As B = 90°


sin2A + sin2B + sin2C = sin2A + sin2(90°) + sin2C


= sin2A + 1 + sin2C


From before, we know that A + C = 90° ; C = 90° - A


sin2A + sin2B + sin2C = sin2A + 1 + sin2( 90° - A)


= sin2A + cos2(A) + 1


[by using the identity cos x = sin ( 90° - x)]


sin2A + sin2B + sin2C = (sin2A + cos2A) + 1


= 1 + 1


= 2


[by using the identity sin2θ + cos2θ = 1]


Therefore, sin2A + sin2B + sin2C = 2.


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