Question

If a line makes angles $ \alpha ,\beta ,\gamma $ and $ \delta $ with four diagonals of a cube, then the value of $ si{{n}^{2}}\alpha +si{{n}^{2}}\beta +si{{n}^{2}}\gamma +si{{n}^{2}}\delta $ is

• A. $ \frac{4}{3} $
• B. $ \frac{8}{3} $
• C. $ \frac{7}{3} $
• D. $ 1 $
• E. None of these

Answer 1

Answer: (B)

We know that, $ {{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma +{{\cos }^{2}}\delta =\frac{4}{3} $ ...(i)
where $ \alpha ,\beta ,\gamma $ and $ \delta $ are the angles with diagonals of cube, then from Eq. (i), we get
$ 1-{{\sin }^{2}}\alpha +1-{{\sin }^{2}}\beta +1-{{\sin }^{2}}\gamma +1-{{\sin }^{2}}\delta =\frac{4}{3} $
$ \Rightarrow $ $ {{\sin }^{2}}\alpha +si{{n}^{2}}\beta +{{\sin }^{2}}\gamma +{{\sin }^{2}}\delta =\frac{8}{3} $