Trigonometric Equations Question 2
Question 2 - 2024 (27 Jan Shift 1)
Let the set of all $a \in R$ such that the equation $\cos 2 x+a \sin x=2 a-7$ 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
Show Answer
Answer (48)
Solution
$\cos 2 x+a \cdot \sin x=2 a-7$
$a(\sin x-2)=2(\sin x-2)(\sin x+2)$
$\sin x=2, \quad a=2(\sin x+2)$
$\Rightarrow a \in[2,6]$
$p=2 \quad q=6$
$r=\tan 9^{\circ}+\cot 9^{\circ}-\tan 27-\cot 27$
$r=\frac{ 1 }{\sin 9 \cdot \cos 9}-\frac{ 1 }{\sin 27 \cdot \cos 27}$
$=2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}\right]$
$r=4$
p.q. . $r=2 \times 6 \times 4=48$
