InterviewSolution
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In a right-angled triangle ABC, write the value of sin2 A + sin2 B + sin2 C. |
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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 sin2A + sin2B + sin2C 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 = (sin 2A + cos 2A) + 1 = 1 + 1 = 2 [by using the identity sin2θ + cos2θ = 1] Therefore, sin2A + sin2B + sin2C = 2. |
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