First slide

Trigonometric equations

Question

General solution of $\sqrt{3} \cos x - \sin x = 1$ is

Moderate

A

$2 n π + \frac{π}{6}, 2 n π - \frac{π}{2}, n \in z$
B

$2 n π + \frac{π}{3}, 2 n π - \frac{π}{3}, n \in z$
C

$2 n π, 2 n π - \frac{2 π}{3}$
D

$2 n π + \frac{π}{4}, 2 n π + \frac{π}{3}$

Solution

$\sqrt{3} \cos x - \sin x = 1$
Dividing by $\sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} = \sqrt{4} = 2$
$\Rightarrow \frac{\sqrt{3}}{2} \cos x - \frac{1}{2} \sin x = \frac{1}{2}$
$\Rightarrow \cos \frac{π}{6} \cos x - \sin \frac{π}{6} \sin x = \frac{1}{2}$
$\Rightarrow \cos (x + \frac{π}{6}) = \cos \frac{π}{3}$
$\Rightarrow x + \frac{π}{6} = 2 n π \pm \frac{π}{3}, n \in z$
$x = 2 n π \pm \frac{π}{3} - \frac{π}{6} = 2 n π + \frac{π}{6}$ or $2 n π - \frac{π}{2}, n \in z$

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