Question

Write the set of values of a for which the equation

\[\sqrt{3} \sin x - \cos x = a\] has no solution.

Answer 1

Given:

\[\sqrt{3} \sin x - \cos x = a\]

\[ \Rightarrow \frac{\sqrt{3} \sin x - \cos x}{2} = \frac{a}{2}\]

\[ \Rightarrow \frac{\sqrt{3}}{2} \sin x - \frac{1}{2} \cos x = \frac{a}{2}\]

\[ \Rightarrow \cos 30^\circ \sin x - \sin 30^\circ \cos x = \frac{a}{2}\]

\[ \Rightarrow \sin ( x - 30^\circ) = \frac{a}{2}\]

\[ \Rightarrow x - 30^\circ = \sin^{- 1} \left( \frac{a}{2} \right)\]

\[ \Rightarrow x = \sin^{- 1} \left( \frac{a}{2} \right) + 30^\circ\]
If \[a = 2\] or \[a = 2\] , then the equation will possess a solution.
For no solution,
\[a \in ( - \infty , - 2) \cup (2, \infty )\].