Question:
A line makes the same angle \(\alpha\) with each of the \(X\) and \(Z\) axes. If the angle \(\beta\) which it makes with the \(Y\)-axis is such that \(\sin ^{2}\beta =2\sin ^{2}\alpha\), then the value of \(\cos ^{2}\alpha\) is ...
A line makes the same angle \(\alpha\) with each of the \(X\) and \(Z\) axes. If the angle \(\beta\) which it makes with the \(Y\)-axis is such that \(\sin ^{2}\beta =2\sin ^{2}\alpha\), then the value of \(\cos ^{2}\alpha\) is ...
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Use that the sum of squares of the direction cosines is 1 together with the given relation.
Updated On: Nov 12, 2026
- \(\frac{1}{2}\)
- \(\frac{1}{4}\)
- \(\frac{3}{5}\)
- \(\frac{4}{5}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept
For any line, \(\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1\). Here the angles with the X and Z axes are both \(\alpha\).
Step 2: Use the relation for beta
\(\sin^{2}\beta=2\sin^{2}\alpha\Rightarrow1-\cos^{2}\beta=2-2\cos^{2}\alpha\).
So \(\cos^{2}\beta=2\cos^{2}\alpha-1\).
Step 3: Use the sum of squares
\(2\cos^{2}\alpha+\cos^{2}\beta=1\Rightarrow2\cos^{2}\alpha+2\cos^{2}\alpha-1=1\).
Step 4: Solve
\(4\cos^{2}\alpha=2\Rightarrow\cos^{2}\alpha=\frac{1}{2}\).
Final Answer:
The value of cos squared alpha is one half. \[ \boxed{\text{(A) }\dfrac{1}{2}} \]
For any line, \(\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1\). Here the angles with the X and Z axes are both \(\alpha\).
Step 2: Use the relation for beta
\(\sin^{2}\beta=2\sin^{2}\alpha\Rightarrow1-\cos^{2}\beta=2-2\cos^{2}\alpha\).
So \(\cos^{2}\beta=2\cos^{2}\alpha-1\).
Step 3: Use the sum of squares
\(2\cos^{2}\alpha+\cos^{2}\beta=1\Rightarrow2\cos^{2}\alpha+2\cos^{2}\alpha-1=1\).
Step 4: Solve
\(4\cos^{2}\alpha=2\Rightarrow\cos^{2}\alpha=\frac{1}{2}\).
Final Answer:
The value of cos squared alpha is one half. \[ \boxed{\text{(A) }\dfrac{1}{2}} \]
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