Question:
If \(\alpha,\beta,\gamma\) are direction angles, then \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma\) is
If \(\alpha,\beta,\gamma\) are direction angles, then \(\sin^2\alpha+\sin^2\beta+\sin^2\gamma\) is
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Always remember: \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\) for direction cosines.
Updated On: Apr 15, 2026
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The Correct Option is B
Solution and Explanation
Concept:
For direction angles:
\[
\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1
\]
Step 1: Use identity. \[ \sin^2\theta = 1 - \cos^2\theta \]
Step 2: Apply to all angles. \[ \sin^2\alpha+\sin^2\beta+\sin^2\gamma = (1-\cos^2\alpha)+(1-\cos^2\beta)+(1-\cos^2\gamma) \] \[ = 3 - (\cos^2\alpha+\cos^2\beta+\cos^2\gamma) \]
Step 3: Substitute value. \[ = 3 - 1 = 2 \]
Step 1: Use identity. \[ \sin^2\theta = 1 - \cos^2\theta \]
Step 2: Apply to all angles. \[ \sin^2\alpha+\sin^2\beta+\sin^2\gamma = (1-\cos^2\alpha)+(1-\cos^2\beta)+(1-\cos^2\gamma) \] \[ = 3 - (\cos^2\alpha+\cos^2\beta+\cos^2\gamma) \]
Step 3: Substitute value. \[ = 3 - 1 = 2 \]
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