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If a line makes angles of measure $\frac{\pi}{6}$ and $\frac{\pi}{3}$ with $\mathrm{X}$ and Y axes respectively, then the angle made by the line with $\mathrm{Z}$ axis is
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Verified Answer
The correct answer is:
$\frac{\pi}{2}$
Let the line make an angle $\theta$ with $Z$ axis.
$$
\begin{array}{l}
\therefore \cos ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}+\cos ^{2} \theta=1 \\
\therefore \cos ^{2} \theta=1-\left(\frac{3}{4}+\frac{1}{4}\right)=0 \Rightarrow \theta=\frac{\pi}{2}
\end{array}
$$
$$
\begin{array}{l}
\therefore \cos ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}+\cos ^{2} \theta=1 \\
\therefore \cos ^{2} \theta=1-\left(\frac{3}{4}+\frac{1}{4}\right)=0 \Rightarrow \theta=\frac{\pi}{2}
\end{array}
$$
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