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If a straight line is equally inclined at an angle $\theta$ with all the three coordinate axes, then $\tan \theta=$
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Verified Answer
The correct answer is:
$\sqrt{2}$
Since the line is making equal angle $\theta$ with the 3 co-ordinate axes.
Hence $\alpha=\beta=\gamma=\theta$
$$
\therefore l=m=n=\cos \theta
$$
We know that, $l^2+m^2+n^2=1$
$$
\begin{aligned}
& \Rightarrow 3 l^2=1 \Rightarrow l=\frac{1}{\sqrt{3}} \\
& \Rightarrow \cos \theta=\frac{1}{\sqrt{3}} \Rightarrow \sin \theta=\frac{\sqrt{2}}{\sqrt{3}}
\end{aligned}
$$
Hence $\tan \theta=\frac{\sin \theta}{\cos \theta}=\sqrt{2}$
Hence $\alpha=\beta=\gamma=\theta$
$$
\therefore l=m=n=\cos \theta
$$
We know that, $l^2+m^2+n^2=1$
$$
\begin{aligned}
& \Rightarrow 3 l^2=1 \Rightarrow l=\frac{1}{\sqrt{3}} \\
& \Rightarrow \cos \theta=\frac{1}{\sqrt{3}} \Rightarrow \sin \theta=\frac{\sqrt{2}}{\sqrt{3}}
\end{aligned}
$$
Hence $\tan \theta=\frac{\sin \theta}{\cos \theta}=\sqrt{2}$
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