Question:
For what value of \( \theta \) lying between \(0\) and \( \pi \) which satisfy inequality \( \sin\theta \cos^3\theta>\sin^3\theta \cos\theta \)
For what value of \( \theta \) lying between \(0\) and \( \pi \) which satisfy inequality \( \sin\theta \cos^3\theta>\sin^3\theta \cos\theta \)
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Convert powers to identities $\longrightarrow$ factor $\longrightarrow$ sign analysis (very powerful method).
Updated On: Apr 22, 2026
- \( \theta \in (\pi/4, \pi/2) \)
- \( \theta \in (0, \pi/4) \)
- \( \theta \in (0, \pi/2) \)
- None of these
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The Correct Option is B
Solution and Explanation
Concept:
\[
\sin\theta \cos^3\theta > \sin^3\theta \cos\theta
\]
Step 1: Simplify.
\[ \sin\theta \cos\theta (\cos^2\theta - \sin^2\theta) > 0 \] \[ \Rightarrow \sin\theta \cos\theta \cos 2\theta > 0 \]
Step 2: Analyze signs.
In \( (0,\pi) \): \[ \sin\theta > 0 \] So condition reduces to: \[ \cos\theta \cos 2\theta > 0 \]
Step 3: Solve.
This is satisfied in: \[ (0, \pi/4) \]
Step 1: Simplify.
\[ \sin\theta \cos\theta (\cos^2\theta - \sin^2\theta) > 0 \] \[ \Rightarrow \sin\theta \cos\theta \cos 2\theta > 0 \]
Step 2: Analyze signs.
In \( (0,\pi) \): \[ \sin\theta > 0 \] So condition reduces to: \[ \cos\theta \cos 2\theta > 0 \]
Step 3: Solve.
This is satisfied in: \[ (0, \pi/4) \]
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