Write the set of values of a for which the equation $\sqrt{3} \sin x-\cos x=a$ has no solution.
Given;
$\sqrt{3} \sin x-\cos x=a$
$\Rightarrow \frac{\sqrt{3} \sin x-\cos x}{2}=\frac{a}{2}$
$\Rightarrow \frac{\sqrt{3}}{2} \sin x-\frac{1}{2} \cos x=\frac{a}{2}$
$\Rightarrow \cos 30^{\circ} \sin x-\sin 30^{\circ} \cos x=\frac{a}{2}$
$\Rightarrow \sin \left(x-30^{\circ}\right)=\frac{a}{2}$
$\Rightarrow x-30^{\circ}=\sin ^{-1}\left(\frac{a}{2}\right)$
$\Rightarrow x=\sin ^{-1}\left(\frac{a}{2}\right)+30^{\circ}$
If $a=2$ or $a=-2$, then the equation will possess a solution.
For no solution, $a \in(-\infty,-2) \cup(2, \infty)$.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.