Question

A line makes acute angles $\alpha ,\beta ,\gamma$ with the coordinate axes such that $\cos \alpha \cos \beta =\cos \beta \cos \gamma =\frac{2}{9}$ and $\cos \gamma \cos \alpha =\frac{4}{9},$ then the value of $\cos \alpha +\cos \beta +\cos \gamma$ is:

• A. $\frac{25}{9}$
• B. $\frac{5}{9}$
• C. $\frac{5}{3}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C $\frac{5}{3}$

$(\cos \alpha +\cos \beta +\cos \gamma {)}^2={\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +2\cos \alpha \cos \beta +2\cos \beta \cos \gamma +2\cos \gamma \cos \alpha =1+\frac{4}{9}+\frac{4}{9}+\frac{8}{9}=\frac{25}{9}$
$\Rightarrow \cos \alpha +\cos \beta +\cos \gamma =\frac{5}{3}$