Trigonometric Equations
Equation with Parameter — Range and Trigonometric Value
nta_pyq_2024_jan
Grade 11
Question:
Let the set of all $a\in\mathbb{R}$ such that the equation $\cos2x+a\sin x=2a-7$ has a solution be $[p,q]$ and $r=\tan9°-\tan27°-\dfrac{1}{\cot63°}+\tan81°$, then $pqr$ is equal to
Step-by-Step Solution
Key Concept: Rewrite: $1-2\sin^2x+a\sin x=2a-7\Rightarrow a(\sin x-2)=2(\sin^2x-4)=2(\sin x-2)(\sin x+2)\Rightarrow a=2(\sin x+2)$ (for $\sin x\neq2$). Since $\sin x\in[-1,1]$: $a\in[2,6]$. So $p=2,q=6$. For $r$: $r=\tan9°-\tan27°-\cot27°+\tan81°=(\tan9°+\cot9°)-(\tan27°+\cot27°)=1/(\sin9°\cos9°)-1/(\sin27°\cos27°)=2/\sin18°-2/\sin54°=4$.
$[p,q]=[2,6]$, $r=4$. $pqr=48$.
Correct Answer: 48