Question

If \( S = \{\theta : \theta \in [-\pi, \pi], \cos\theta \cos(50^\circ/2) - \cos 70^\circ \cos(70^\circ/2) = 0\ \), then \( n(S) \) is equal to:

• A. 17
• B. 19
• C. 21
• D. 23

Hint:

Always check the coefficient of \( \theta \). A function \( \cos(k\theta) \) will have \( 2k \) solutions in a \( 2\pi \) interval.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The equation involves products of cosines. We use the product-to-sum formula: \( 2 \cos A \cos B = \cos(A + B) + \cos(A - B) \).
Step 2: Key Formula or Approach:
Multiply both sides by 2:
\[ 2 \cos \theta \cos \frac{5\theta}{2} = 2 \cos 7\theta \cos \frac{7\theta}{2} \]
\[ \cos \frac{7\theta}{2} + \cos \frac{3\theta}{2} = \cos \frac{21\theta}{2} + \cos \frac{7\theta}{2} \]
Step 3: Detailed Explanation:
Canceling \( \cos \frac{7\theta}{2} \) from both sides:
\[ \cos \frac{21\theta}{2} - \cos \frac{3\theta}{2} = 0 \]
Use the sum-to-product formula \( \cos C - \cos D = -2 \sin \frac{C+D}{2} \sin \frac{C-D}{2} \):
\[ -2 \sin \left( \frac{\frac{21\theta}{2} + \frac{3\theta}{2}}{2} \right) \sin \left( \frac{\frac{21\theta}{2} - \frac{3\theta}{2}}{2} \right) = 0 \]
\[ -2 \sin (6\theta) \sin \left( \frac{9\theta}{2} \right) = 0 \]
This implies either \( \sin 6\theta = 0 \) or \( \sin \frac{9\theta}{2} = 0 \).
Case 1: \( \sin 6\theta = 0 \implies 6\theta = n\pi \implies \theta = \frac{n\pi}{6} \).
For \( \theta \in [-\pi, \pi] \), \( n \in \{-6, -5, \dots, 0, \dots, 5, 6\} \). This gives 13 solutions.
Case 2: \( \sin \frac{9\theta}{2} = 0 \implies \frac{9\theta}{2} = m\pi \implies \theta = \frac{2m\pi}{9} \).
For \( \theta \in [-\pi, \pi] \), \( m \in \{-4, -3, \dots, 0, \dots, 3, 4\} \).
Values of \( \theta \) are \( \{ 0, \pm \frac{2\pi}{9}, \pm \frac{4\pi}{9}, \pm \frac{6\pi}{9}, \pm \frac{8\pi}{9} \} \).
Note: \( \theta = 0 \) and \( \pm \frac{6\pi}{9} = \pm \frac{2\pi}{3} \) are already counted in Case 1.
Additional solutions from Case 2 are \( \pm \frac{2\pi}{9}, \pm \frac{4\pi}{9}, \pm \frac{8\pi}{9} \), which are 6 more solutions.
Total solutions = 13 + 6 = 19.
Step 4: Final Answer:
The number of elements in set S is 19.