A line makes the same angle ' $\alpha$ ' with each of the $\mathrm{x}$ and $\mathrm{y}$ axes. If the angle ' $ heta$ ', which it makes with the z-axis, is such that $\sin ^2 heta=2 \sin ^2 \alpha$, then the angle $\alpha$ is

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A line makes the same angle ' $\alpha$ ' with each of the $\mathrm{x}$ and $\mathrm{y}$ axes. If the angle ' $	heta$ ', which it makes with the z-axis, is such that $\sin ^2 	heta=2 \sin ^2 \alpha$, then the angle $\alpha$ is

Aicharya · Accepted Answer

Direction cosines of the line are $\cos ^2 \alpha+\cos ^2 \alpha+\cos ^2 	heta=1$ $\begin{aligned} & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-\sin ^2 	heta=1 \ & \Rightarrow \cos ^2 \alpha+\cos ^2 \alpha+1-2 \sin ^2 \alpha=1 \ & \Rightarrow 1-\sin ^2 \alpha+1-\sin ^2 \alpha+1-2 \sin ^2 \alpha=1 \ & \Rightarrow \sin ^2 \alpha=\frac{1}{2}=\sin ^2 \frac{\pi}{4} \ & \Rightarrow \alpha=\frac{\pi}{4}\end{aligned}$

A line makes the same angle ' $\alpha$ ' with each of the $\mathrm{x}$ and $\mathrm{y}$ axes. If the angle '…

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