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:

$$\begin{gathered} \sqrt{3}\sin x-\cos x=a \\ \Rightarrow \frac{\sqrt{3}\sin x-\cos x}{2}=\frac{a}{2} \\ \Rightarrow \frac{\sqrt{3}}{2}\sin x-\frac{1}{2}\cos x=\frac{a}{2} \\ \Rightarrow \cos 30°\sin x-\sin 30°\cos x=\frac{a}{2} \\ \Rightarrow \sin (x-30°)=\frac{a}{2} \\ \Rightarrow x-30°={\sin }^{-1}\left(\frac{a}{2}\right) \\ \Rightarrow x={\sin }^{-1}\left(\frac{a}{2}\right)+30° \end{gathered}$$

If $a=2$ or $a=-2$ , then the equation will possess a solution.
For no solution, $a\in (-\infty ,-2)\cup (2,\infty )$.