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,

Question

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 $r=	an 9^{\circ}-	an 27^{\circ}-\frac{1}{\cot 63^{\circ}}+	an 81^{\circ}$, then pqr is equal to $\_\_\_\_$ .\

Professor PC Thomas Classes · Answer

(48): $\cos 2 x+a \sin x=2 a-7$\ $1-2 \sin ^{2} x+a \sin x=2 a-7$\ Let $t=\sin x 	herefore 2 t^{2}-a t+2 a-8=0$\ $\Rightarrow \quad(t-2)(2 t+4-a)=0$\ $\Rightarrow \quad t=2$ or $t=\frac{a-4}{2}$\ $	herefore \quad \sin x=2$ (not possible) or $2 \sin x=a-4$\ $\Rightarrow \quad a=2 \sin x+4 \in[2,6][\because \sin x \in[-1,1]]$\ So, $p=2, q=6$\ Now, $r=	an 9^{\circ}-	an 27^{\circ}-\frac{1}{\cot 63^{\circ}}+	an 81^{\circ}$ $$ \begin{aligned} & =\left(	an 9^{\circ}+\cot 9^{\circ}ight)-\left(	an 27^{\circ}+\cot 27^{\circ}ight) \ & =2 \operatorname{cosec} 18^{\circ}-2 \operatorname{cosec} 54^{\circ} \ & \quad[\because 	an 	heta+\cot 	heta=2 \operatorname{cosec} 2 	heta] \ & =2\left[\frac{4}{\sqrt{5}-1}-\frac{4}{\sqrt{5}+1}ight]=4 \Rightarrow r=4 \ 	herefore \quad & p q r=2 	imes 6 	imes 4=48 \end{aligned} $$