Step 1: Understanding the Concept:
We need to solve a trigonometric inequality within the interval \( (0, \pi) \). Factorization and using double-angle identities are effective strategies here.
Step 2: Key Formula or Approach:
1. Use the identity \( \sin(2\theta) = 2 \sin\theta \cos\theta \).
2. Use the identity \( \cos(2\theta) = \cos^2\theta - \sin^2\theta \).
Step 3: Detailed Explanation:
The given inequality is:
\[ \sin\theta \cos^3 \theta>\sin^3 \theta \cos\theta \]
Bring all terms to one side:
\[ \sin\theta \cos^3 \theta - \sin^3 \theta \cos\theta>0 \]
Factor out the common term \( \sin\theta \cos\theta \):
\[ \sin\theta \cos\theta (\cos^2 \theta - \sin^2 \theta)>0 \]
Substitute the identities:
\[ \frac{1}{2} \sin(2\theta) \cos(2\theta)>0 \]
\[ \sin(4\theta)>0 \]
For \( \sin(4\theta)>0 \), the angle \( 4\theta \) must be in the first or second quadrant:
\[ 0<4\theta<\pi \]
Dividing by 4:
\[ 0<\theta<\frac{\pi}{4} \]
Thus, \( \theta \in (0, \pi/4) \) satisfies the inequality.
Step 4: Final Answer:
The value of \( \theta \) lies in the interval \( (0, \pi/4) \).