Write the set of values of a for which the equation

Question:

Write the set of values of a for which the equation $\sqrt{3} \sin x-\cos x=a$ has no solution.

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, $a \in(-\infty,-2) \cup(2, \infty)$.

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