Question:
A line in the 3-dimensional space makes an angle \(\theta\left(0 \lt \theta \le \frac\pi2\right)\) with both the \(X\) and \(Y\)-axes. Then, the set of all values of \(\theta\) is in the interval :
A line in the 3-dimensional space makes an angle \(\theta\left(0 \lt \theta \le \frac\pi2\right)\) with both the \(X\) and \(Y\)-axes. Then, the set of all values of \(\theta\) is in the interval :
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Use \(l^2 + m^2 + n^2 = 1\) with \(l = m = \cos\theta\) and require \(n^2 \ge 0\).
Updated On: Nov 11, 2026
- \(\left(\frac\pi3, \frac\pi2\right]\)
- \(\left(0, \frac\pi4\right]\)
- \(\left[\frac\pi4, \frac\pi2\right]\)
- \(\left[\frac\pi6, \frac\pi3\right]\)
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The Correct Option is C
Solution and Explanation
Let the line make angle \(\gamma\) with the \(Z\)-axis. Direction cosines satisfy \(\cos^2\theta + \cos^2\theta + \cos^2\gamma = 1\).
So \(\cos^2\gamma = 1 - 2\cos^2\theta \ge 0\), giving \(\cos^2\theta \le \frac12\).
With \(0 \lt \theta \le \frac\pi2\), \(\cos\theta \le \frac{1}{\sqrt2}\) means \(\theta \ge \frac\pi4\).
\(\theta = \frac\pi2\) is attainable (the Z-axis itself). So \(\theta \in \left[\frac\pi4, \frac\pi2\right]\).
Answer: (3) \(\left[\frac\pi4, \frac\pi2\right]\)
So \(\cos^2\gamma = 1 - 2\cos^2\theta \ge 0\), giving \(\cos^2\theta \le \frac12\).
With \(0 \lt \theta \le \frac\pi2\), \(\cos\theta \le \frac{1}{\sqrt2}\) means \(\theta \ge \frac\pi4\).
\(\theta = \frac\pi2\) is attainable (the Z-axis itself). So \(\theta \in \left[\frac\pi4, \frac\pi2\right]\).
Answer: (3) \(\left[\frac\pi4, \frac\pi2\right]\)
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