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(C) Let the direction cosines of the line be $l, m, n$. Since the line makes an angle $	heta$ with the $x$ and $y$ axes,we have $l = \cos 	heta$ and

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$\left[ \frac{\pi}{4}, \frac{\pi}{2} \right]$

Vedclass · Answer

(C) Let the direction cosines of the line be $l, m, n$. Since the line makes an angle $	heta$ with the $x$ and $y$ axes,we have $l = \cos 	heta$ and $m = \cos 	heta$.Let the angle with the $z$-axis be $\phi$. Then $n = \cos \phi$.The condition for direction cosines is $l^2 + m^2 + n^2 = 1$.Substituting the values,we get $\cos^2 	heta + \cos^2 	heta + \cos^2 \phi = 1$,which implies $2 \cos^2 	heta + \cos^2 \phi = 1$.Thus,$\cos^2 \phi = 1 - 2 \cos^2 	heta = - \cos 2 	heta$.Since $\cos^2 \phi \ge 0$,we must have $-\cos 2 	heta \ge 0$,which implies $\cos 2 	heta \le 0$.Given $0 For $\cos 2 	heta \le 0$ in the interval $(0, \pi]$,we have $\frac{\pi}{2} \le 2 	heta \le \pi$.Dividing by $2$,we get $\frac{\pi}{4} \le 	heta \le \frac{\pi}{2}$.Therefore,the set of all values of $	heta$ is $\left[ \frac{\pi}{4}, \frac{\pi}{2} ight]$.

$A$ line in the $3-$dimensional space makes an angle $\theta \left( 0 < \theta \le \frac{\pi}{2} \right)$ with both the $x$ and $y$ axes. Then the set of all values of $\theta$ is the interval

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