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Q.
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
TS EAMCET 2021
Solution:
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 $\cos \frac{\pi}{4}, \cos \frac{\pi}{3}, \cos \frac{\pi}{3}$
$\Rightarrow \frac{1}{\sqrt{2}}, \frac{1}{2}, \frac{1}{2}$