First slide

Trigonometric equations

Question

The least positive solution of $\cot (\frac{π}{3 \sqrt{3}} \sin 2 x) = \sqrt{3}$ lies in

Moderate

A

$(0, \frac{π}{6}]$
B

$(\frac{π}{9}, \frac{π}{6})$
C

$(\frac{π}{12}, \frac{π}{9}]$
D

$(\frac{π}{3}, \frac{π}{2}]$

Solution

$\begin{array}{l} \cot (\frac{π}{3 \sqrt{3}} \sin 2 x) = \sqrt{3} \\ \Rightarrow \frac{π}{3 \sqrt{3}} \sin 2 x = nπ + \frac{π}{6}, n \in Z \\ \Rightarrow \sin 2 x = 3 \sqrt{3} n + \frac{\sqrt{3}}{2} \end{array}$
for solution n= 0
$\begin{array}{l} \Rightarrow \sin 2 x = \frac{\sqrt{3}}{2} \\ \Rightarrow 2 x = \frac{π}{3} \Rightarrow x = \frac{π}{6} \end{array}$

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