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Giải bài 56 trang 30 sách bài tập toán 11 - Cánh diều

Phương trình \(\cos 2x = \cos \left( {x + \frac{\pi }{4}} \right)\) có các nghiệm là:

꧂Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Phương trình \(\cos 2x = \cos \left( {x + \frac{\pi }{4}} \right)\) có các nghiệm là: A. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x =  - \frac{\pi }{4} + k\frac{{2\pi }}{3}}\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)            B. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x =  - \frac{\pi }{{12}} + k\frac{{2\pi }}{3}}\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\) C. \(\left[ {\begin{array}{*{20}{c}}{x =  - \frac{\pi }{4} + k2\pi }\\{x =  - \frac{\pi }{{12}} + k\frac{{2\pi }}{3}}\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)                             D. \(\left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{4} + k2\pi }\\{x =  - \frac{\pi }{{12}} + k2\pi }\end{array}{\rm{ }}\left( {k \in \mathbb{Z}} \right)} \right.\)

Phương pháp giải - Xem chi tiết

Sử dụng kết quả \(\cos x = \cos \alpha  \Leftrightarrow \left[ \begin{array}{l}x = \alpha  + k2\pi \\x =  - \alpha  + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)

Lời giải chi tiết

Ta có:\(\cos 2x = \cos \left( {x + \frac{\pi }{4}} \right) \Leftrightarrow \left[ \begin{array}{l}2x = x + \frac{\pi }{4} + k2\pi \\2x =  - x - \frac{\pi }{4} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{4} + k2\pi \\3x =  - \frac{\pi }{4} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{4} + k2\pi \\x =  - \frac{\pi }{{12}} + k\frac{{2\pi }}{3}\end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)Đáp án đúng là B.

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