Giải phương trình: \(3{\sin ^2}x + 2{\cos ^4}x - 2...

B

- Hướng dẫn giải

Phương pháp giải:

Giải chi tiết:

\(\begin{array}{l}3{\sin ^2}x + 2{\cos ^4}x - 2 = 0 \Leftrightarrow 3\left( {1 - {{\cos }^2}x} \right) + 2{\cos ^4}x - 2 = 0\\ \Leftrightarrow 3 - 3{\cos ^2}x + 2{\cos ^4}x - 2 = 0 \Leftrightarrow 2{\cos ^4}x - 3{\cos ^2}x + 1 = 0\left(  *  \right)\end{array}\)

Đặt \({\cos ^2} = t\,\,\,\left( {0 \le t \le 1} \right)\)\( \Rightarrow \left(  *  \right) \Leftrightarrow 2{t^2} - 3t + 1 = 0\)

\(\begin{array}{l} \Rightarrow \left[ \begin{array}{l}t = 1\\t = \dfrac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}{\cos ^2}x = 1\\{\cos ^2}x = \dfrac{1}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\left[ \begin{array}{l}\cos x = 1\\\cos x =  - 1\end{array} \right.\\\left[ \begin{array}{l}\cos x =  - \dfrac{1}{{\sqrt 2 }}\\\cos x = \dfrac{1}{{\sqrt 2 }}\end{array} \right.\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}\left[ \begin{array}{l}\cos x = \cos \left( 0 \right)\\\cos x = \cos \left( \pi  \right)\end{array} \right.\\\left[ \begin{array}{l}\cos x = \cos \left( {\dfrac{{3\pi }}{4}} \right)\\\cos x = \cos \left( {\dfrac{\pi }{4}} \right)\end{array} \right.\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\left[ \begin{array}{l}x = k2\pi \\x = \pi  + k2\pi \end{array} \right.\\\left[ \begin{array}{l}x =  \pm \dfrac{{3\pi }}{4} + k2\pi \\x =  \pm \dfrac{\pi }{4} + k2\pi \end{array} \right.\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = k\pi \\x = \dfrac{\pi }{4} + \dfrac{{k\pi }}{2}\end{array} \right.\,\,\,\left( {k \in \mathbb{Z}} \right)\end{array}\)

Vậy \(S = \left\{ {k\pi ;\dfrac{\pi }{4} + \dfrac{{k\pi }}{2}} \right\}\)