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
$1-{\sin }^2\alpha +1-{\sin }^2\beta +1-{\sin }^2\gamma +1-{\sin }^2\delta =\frac{4}{3}$
$\Rightarrow$ ${\sin }^2\alpha +si{n}^2\beta +{\sin }^2\gamma +{\sin }^2\delta =\frac{8}{3}$