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

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:
$p · q · r = 2 · 6 · 4 = 48$