Question

If $ \alpha ,\beta ,\gamma $ are the angles when a directed line makes with the positive directions the coordinate axes, then $ {{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma $ is equal to:

• A. 1
• B. 2
• C. 3
• D. none of these

Answer 1

Answer: (B)

If the line makes an angle $\alpha, \beta, \gamma$ to the coordinate axes, then
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
We know that $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$
$\Rightarrow 1-\sin ^{2} \alpha+1-\sin ^{2} \beta+1-\sin ^{2} \gamma=1$
$\Rightarrow \sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma=3-1$
$=2$