Question

The direction cosines of the line making angles $\frac{\pi }{4},\frac{\pi }{3}$ and $\theta \left(0<\theta <\frac{\pi }{2}\right)$ respectively with $x,y$ and $z$ axes, are

• A. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{\sqrt{3}}{2}$
• C. $\frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{\sqrt{2}}$
• D. $\frac{1}{\sqrt{2}},\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

Let $\alpha =\frac{\pi }{4},\beta =\frac{\pi }{3}$ and $\gamma =\theta$
Then, ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
or ${\sin }^2\gamma ={\cos }^2\alpha +{\cos }^2\beta$
$\Rightarrow {\sin }^2\theta ={\cos }^2\frac{\pi }{4}+{\cos }^2\frac{\pi }{3}$
$=\frac{1}{2}+\frac{1}{4}=\frac{3}{4}$
$\Rightarrow \sin \theta =\frac{\sqrt{3}}{2}=\sin \frac{\pi }{3}$
$\Rightarrow \theta =\frac{\pi }{3}$
Direction cosines are $\cos \frac{\pi }{4},\cos \frac{\pi }{3},\cos \frac{\pi }{3}$
$\Rightarrow \frac{1}{\sqrt{2}},\frac{1}{2},\frac{1}{2}$