Question

If a line makes angles $\alpha ,\beta ,\gamma$ with coordinate axes, Find the value of ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma$

Answer 1

If $\alpha ,\beta ,\gamma$ is the angle, a line makes with the co-ordinate axis,

$\therefore {\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

$\Rightarrow (1-{\sin }^2\alpha )+(1-{\sin }^2\beta )+(1-{\sin }^2\gamma )=1$

$\Rightarrow 3-({\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma )=1$

$\therefore ({\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma )=2$