If \( \cos \theta = \frac{1}{3} \), then what is the value of \( \sin \left( \frac{\theta}{2} \right) \sin \left( \frac{3\theta}{2} \right) \) ?
- A. \frac{5}{9} ✓
- B. \frac{7}{9}
- C. \frac{10}{9}
- D. \frac{11}{9}
Correct Answer: A. \frac{5}{9}
Explanation
Using the product to sum identity, the expression equals \frac{1}{2}(\cos\theta - \cos 2\theta). Substituting \cos\theta = 1/3 and \cos 2\theta = 2(1/3)^2 - 1 = -7/9 yields \frac{1}{2}(1/3 + 7/9) = 5/9.