Question

If a line makes angles $\alpha ,\beta ,\gamma$ with positive axes, then the range of $\sin \alpha \sin \beta +\sin \beta \sin \gamma +\sin \gamma \sin \alpha$ is

• A. $(\frac{-1}{2},1)$
• B. $(\frac{1}{2},2)$
• C. $(-1,2)$
• D. $(-1,2]$

Answer 1

Answer: (D)

$(-1,2]$

If a line makes an angle $\alpha ,\beta ,\gamma$ with the axes, then

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ $[\because \cos \alpha ,\cos \beta ,\cos \gamma$ are directional cosines$]$

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

$\Rightarrow {\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$

Also ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma \ge \sin \alpha \sin \beta +\sin \beta \sin \gamma +\sin \gamma \sin \alpha$

$\Rightarrow \sum \sin \alpha \sin \beta \le 2$ $\dots \left(1\right)$

Also $(\sum \sin \alpha {)}^2=\sum {\sin }^2\alpha +2\sum \sin \alpha \sin \beta$ $[\because (a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca]$

$(\sum \sin \alpha {)}^2>0$

$\sum {\sin }^2\alpha +2\sum \sin \alpha \sin \beta >0$

$2+2\sum \sin \alpha \sin \beta >0$

$2\sum \sin \alpha \sin \beta >-2$

$\sum \sin \alpha \sin \beta >\frac{-2}{2}$

$\sum \sin \alpha \sin \beta >-1$ $\dots \left(2\right)$

From $\left(1\right)and\left(2\right)$

$2\ge \sum \sin \alpha \sin \beta >-1$

Hence the range of $\sum \sin \alpha \sin \beta$ is $(-1,2]$