Question:
If a line makes angles \(α,β,γ\) with the coordinate axes, then the sum of values of \(sin^2α+sin^2β+sin^2γ\) and \(cos2α+cos2β+cos2γ\) is ...
If a line makes angles \(α,β,γ\) with the coordinate axes, then the sum of values of \(sin^2α+sin^2β+sin^2γ\) and \(cos2α+cos2β+cos2γ\) is ...
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Direction cosines satisfy \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\).
Updated On: Oct 1, 2026
- \(5\)
- \(0\)
- \(3\)
- \(1\)
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The Correct Option is D
Solution and Explanation
Step 1: Key Fact:
For a line making angles \(\alpha,\beta,\gamma\) with the axes, \(l^2+m^2+n^2 = 1\), i.e. \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\).
Step 2: First sum:
\(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 3 - 1 = 2\).
Step 3: Second sum:
\(\cos2\alpha + \cos2\beta + \cos2\gamma = 2(\cos^2\alpha+\cos^2\beta+\cos^2\gamma) - 3 = 2 - 3 = -1\).
Step 4: Add:
\(2 + (-1) = 1\).
Final Answer:
The required sum is \(1\), option (D). \[ \boxed{1} \]
For a line making angles \(\alpha,\beta,\gamma\) with the axes, \(l^2+m^2+n^2 = 1\), i.e. \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\).
Step 2: First sum:
\(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 3 - 1 = 2\).
Step 3: Second sum:
\(\cos2\alpha + \cos2\beta + \cos2\gamma = 2(\cos^2\alpha+\cos^2\beta+\cos^2\gamma) - 3 = 2 - 3 = -1\).
Step 4: Add:
\(2 + (-1) = 1\).
Final Answer:
The required sum is \(1\), option (D). \[ \boxed{1} \]
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