Question

If a line makes angles $\alpha ,\beta$ and $\gamma$ with three-dimensional coordinate axes, respectively, then find the value of $\cos 2\alpha +\cos 2\beta +\cos 2\gamma$.

Answer 1

[a]l$=cos\alpha$

$m=cos\beta$

$n=cos\gamma$

we now ,

$l^2+m^2+n^2=1$

$co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$

$weknowco{s}^2\theta =2co{s}^2\theta -1$

so,

$\frac{(}{1-co{s}^2\alpha }2)+(\frac{1-co{s}^2\beta }{2})+(\frac{1-co{s}^2\gamma }{2})=1$

$=(1-co{s}^2\alpha )+(1-co{s}^2\beta )+(1-co{s}^2\gamma )=2$

$=-1=co{s}^2=co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma$

Hence value is -1