MHT CET · Maths · Three Dimensional Geometry
If \(\alpha, \beta\) and \(\gamma\) are the angles which a half ray makes with the positive direction of the axes, then \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\) is equal to
- A 1
- B 2
- C 0
- D \(-1\)
Answer & Solution
Correct Answer
(B) 2
Step-by-step Solution
Detailed explanation
Given expression, \(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\)
\(\begin{array}{r}
=\left(1-\cos ^{2} \alpha\right)+\left(1-\cos ^{2} \beta\right)+\left(1-\cos ^{2} \gamma\right) \\
=3-\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=3-1=2 \\
\left(\because \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\right)\end{array}\)
\(\begin{array}{r}
=\left(1-\cos ^{2} \alpha\right)+\left(1-\cos ^{2} \beta\right)+\left(1-\cos ^{2} \gamma\right) \\
=3-\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=3-1=2 \\
\left(\because \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\right)\end{array}\)
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