Phương trình:  \({\sin ^7}x + {\cos ^5}x + \frac{1...

C

- Hướng dẫn giải

Phương pháp giải:

-        Nhóm để được nhân tử chung \(\left( {\sin x + \cos x} \right)\)

\(\begin{array}{l} \Leftrightarrow {\sin ^7}x + {\cos ^5}x + ({\sin ^5}x + {\cos ^3}x)\sin x\cos x = \sin x + \cos x\\ \Leftrightarrow {\sin ^7}x + {\cos ^5}x + {\sin ^6}x\cos x + \sin x{\cos ^4}x = \sin x + \cos x\\ \Leftrightarrow ({\sin ^7}x + {\sin ^6}x\cos x) + ({\cos ^5}x + \sin x{\cos ^4}x) = \sin x + \cos x\\ \Leftrightarrow {\sin ^6}x(\sin x + \cos x) + {\cos ^4}x(\sin x + \cos x) = \sin x + \cos x\end{array}\)

Giải chi tiết:

\(\begin{array}{l}Pt \Leftrightarrow {\sin ^7}x + {\cos ^5}x + ({\sin ^5}x + {\cos ^3}x)\sin x\cos x = \sin x + \cos x\\ \Leftrightarrow {\sin ^7}x + {\cos ^5}x + {\sin ^6}x\cos x + \sin x{\cos ^4}x = \sin x + \cos x\\ \Leftrightarrow ({\sin ^7}x + {\sin ^6}x\cos x) + ({\cos ^5}x + \sin x{\cos ^4}x) = \sin x + \cos x\\ \Leftrightarrow {\sin ^6}x(\sin x + \cos x) + {\cos ^4}x(\sin x + \cos x) = \sin x + \cos x\\ \Leftrightarrow (\sin x + \cos x)({\sin ^6}x + {\cos ^4}x - 1) = 0 \Leftrightarrow \left[ \begin{array}{l}\sin x + \cos x = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\{\sin ^6}x + {\cos ^4}x = 1\,\,\,\,\,\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}\sin \left( {x + \frac{\pi }{4}} \right) = 0\\{\sin ^6}x + {\left( {1 - {{\sin }^2}x} \right)^2} = 1\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x + \frac{\pi }{4} = k\pi \\{\sin ^6}x + {\sin ^4}x - 2{\sin ^2}x = 0\end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{4} + k\pi \\{\sin ^2}x\left( {{{\sin }^4}x + {{\sin }^2}x - 2} \right) = 0\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{4} + k\pi \\\sin x = 0\\{\sin ^2}x = 1 \Leftrightarrow \cos x = 0\\{\sin ^2}x =  - 2\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x =  - \frac{\pi }{4} + k\pi \\x = m\pi \\x = \frac{\pi }{2} + l\pi \end{array} \right.\;\;\left( {k,\;m,\;l \in Z} \right)\end{array}\)

Vậy phương trình có nghiệm thuộc \(\left[ {0;\;10\pi } \right]\)

\( \Leftrightarrow \left[ \begin{array}{l}0 \le  - \frac{\pi }{4} + k\pi  \le 10\pi \\0 \le m\pi  \le 10\pi \\0 \le \frac{\pi }{2} + l\pi  \le 10\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\frac{1}{4} \le k \le \frac{{41}}{4}\\0 \le m \le 10\\ - \frac{1}{2} \le l \le \frac{{19}}{2}\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}k \in \left\{ {1;\;2;\;3;......;\;9;\;10} \right\}\\m \in \left\{ {0;\;1;\;2;......;\;9;\;10} \right\}\\l \in \left\{ {0;\;1;.......;\;9} \right\}\end{array} \right.\)

Có 31 nghiệm thõa mãn bài toán.