Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Write the Set of Values of a for Which the Equation √ 3 Sin X − Cos X = a Has No Solution. - Mathematics

Sum

Write the set of values of a for which the equation

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

#### Solution

Given:

$\sqrt{3} \sin x - \cos x = a$

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

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

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

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

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

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

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Q 4 | Page 26