Question

Let the set of all \( a \in \mathbb{R} \) such that the equation \(\cos 2x + a \sin x = 2a - 7\) has a solution be \([p, q]\) and \( r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ + \tan 81^\circ} \), then \( pqr \) is equal to ______.

Answer 1

Correct Answer: 48

Given the equation:  
\( \cos 2x + a \sin x = 2a - 7 \)  

We need to find the set of all \( a \in \mathbb{R} \) such that this equation has a solution in the interval \( [p, q] \), and find the value of \( pqr \) where:  
\( r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ + \tan 81^\circ} \)  

Step 1. Analyzing the Equation: Rewrite the equation as:  
  \( a(\sin x - 2) = 2(\sin x - 2)(\sin x + 2) \)

  For \( \sin x = 2 \), we have:  
  \( a = 2(\sin x + 2) \)

  Therefore, the values of \( a \) lie in the interval:  
  \( a \in [2, 6] \)

  So, \( p = 2 \) and \( q = 6 \).

Step 2. Calculating \( r \): Given:  
  \( r = \tan 9^\circ - \tan 27^\circ - \frac{1}{\cot 63^\circ + \tan 81^\circ} \)

  Using trigonometric identities:  
  \( \cot 63^\circ + \tan 81^\circ = \frac{1}{\tan 27^\circ + \tan 81^\circ} \)

Simplifying further:
\(r = 4\)

Step 3. Calculating pqr:
\(p · q · r = 2 · 6 · 4 = 48\)