Giải phương trình \(3\sin x + 2\cos x = 2 + 3\tan...

D

- Hướng dẫn giải

Lời giải chi tiết:

\(3\sin x + 2\cos x = 2 + 3\tan x\,\,\,\left( 1 \right)\).

Điều kiện: \(\cos x \ne 0 \Leftrightarrow x \ne \frac{\pi }{2} + k\pi \,\,\left( {k \in \mathbb{Z}} \right)\).

\(\begin{array}{l}\left( 1 \right)\,\,\, \Leftrightarrow 3\sin x\cos x + 2{\cos ^2}x = 2\cos x + 3\sin x\\ \Leftrightarrow 3\sin x\cos x - 3\sin x + 2{\cos ^2}x - 2\cos x = 0\\ \Leftrightarrow 3\sin x\left( {\cos x - 1} \right) + 2\cos x\left( {\cos x - 1} \right) = 0\\ \Leftrightarrow \left( {3\sin x + 2\cos x} \right)\left( {\cos x - 1} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l}3\sin x + 2\cos x = 0\\\cos x - 1 = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}3\tan x + 2 = 0\\\cos x = 1\end{array} \right.\,\,\,\,\\ \Leftrightarrow \left[ \begin{array}{l}\tan x = \frac{{ - 2}}{3}\\\cos x = 1\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \arctan \frac{{ - 2}}{3} + k\pi     \left( {tm} \right)\\x = k2\pi \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {tm} \right)\end{array} \right.\end{array}\)

Vậy \(S = \left\{ {k2\pi ,\,\,\arctan \frac{2}{3} + k\pi ,\,\,k \in \mathbb{Z}} \right\}\).

Chọn D.