Đề bài
Giải phương trình:
a) \(\sin \left( {2x + \frac{\pi }{3}} \right) = \sin \left( {3x - \frac{\pi }{6}} \right)\)
b) \(\cos \left( {x + \frac{\pi }{4}} \right) = \cos \left( {\frac{\pi }{4} - 2x} \right)\)
c) \({\cos ^2}\left( {\frac{x}{2} + \frac{\pi }{6}} \right) = {\cos ^2}\left( {\frac{{3x}}{2} + \frac{\pi }{4}} \right)\)
d) \(\cot 3x = \tan \frac{{2\pi }}{7}\)
Phương pháp giải - Xem chi tiết
a) Sử dụng kết quả \(\sin x = \sin \alpha \Leftrightarrow \left[ \begin{array}{l}x = \alpha + k2\pi \\x = \pi - \alpha + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
b) Sử dụng kết quả \(\cos x = \cos \alpha \Leftrightarrow \left[ \begin{array}{l}x = \alpha + k2\pi \\x = - \alpha + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
c) Sử dụng công thức \({\cos ^2}x = \frac{{1 + \cos 2x}}{2}\)
Sử dụng kết quả \(\cos x = \cos \alpha \Leftrightarrow \left[ \begin{array}{l}x = \alpha + k2\pi \\x = - \alpha + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
d) Sử dụng công thức \(\tan x = \cot \left( {\frac{\pi }{2} - x} \right)\) và kết quả \(\cot x = \cot \alpha \Leftrightarrow x = \alpha + k\pi \)\(\left( {k \in \mathbb{Z}} \right)\)
Lời giải chi tiết
a) Ta có:
\(\sin \left( {2x + \frac{\pi }{3}} \right) = \sin \left( {3x - \frac{\pi }{6}} \right) \Leftrightarrow \left[ \begin{array}{l}2x + \frac{\pi }{3} = 3x - \frac{\pi }{6} + k2\pi \\2x + \frac{\pi }{3} = \pi - 3x + \frac{\pi }{6} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} - x = - \frac{\pi }{2} + k2\pi \\5x = \frac{{5\pi }}{6} + k2\pi \end{array} \right.\)
\( \Leftrightarrow \left[ \begin{array}{l}x = \frac{\pi }{2} + k2\pi \\x = \frac{\pi }{6} + k\frac{{2\pi }}{5}\end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
b) Ta có:
\(\cos \left( {x + \frac{\pi }{4}} \right) = \cos \left( {\frac{\pi }{4} - 2x} \right) \Leftrightarrow \left[ \begin{array}{l}x + \frac{\pi }{4} = \frac{\pi }{4} - 2x + k2\pi \\x + \frac{\pi }{4} = 2x - \frac{\pi }{4} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}3x = k2\pi \\ - x = - \frac{\pi }{2} + k2\pi \end{array} \right.\)
\( \Leftrightarrow \left[ \begin{array}{l}x = k\frac{{2\pi }}{3}\\x = \frac{\pi }{2} + k2\pi \end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
c) Ta có:
\({\cos ^2}\left( {\frac{x}{2} + \frac{\pi }{6}} \right) = \frac{{1 + \cos \left[ {2\left( {\frac{x}{2} + \frac{\pi }{6}} \right)} \right]}}{2} = \frac{{1 + \cos \left( {x + \frac{\pi }{3}} \right)}}{2}\);
\({\cos ^2}\left( {\frac{{3x}}{2} + \frac{\pi }{4}} \right) = \frac{{1 + \cos \left[ {2\left( {\frac{{3x}}{2} + \frac{\pi }{4}} \right)} \right]}}{2} = \frac{{1 + \cos \left( {3x + \frac{\pi }{2}} \right)}}{2}\)
Phương trình trở thành:
\(\frac{{1 + \cos \left( {x + \frac{\pi }{3}} \right)}}{2} = \frac{{1 + \cos \left( {3x + \frac{\pi }{2}} \right)}}{2} \Leftrightarrow \cos \left( {x + \frac{\pi }{3}} \right) = \cos \left( {3x + \frac{\pi }{2}} \right)\)
\( \Leftrightarrow \left[ \begin{array}{l}x + \frac{\pi }{3} = 3x + \frac{\pi }{2} + k2\pi \\x + \frac{\pi }{3} = - 3x - \frac{\pi }{2} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} - 2x = \frac{\pi }{6} + k2\pi \\4x = - \frac{{5\pi }}{6} + k2\pi \end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = - \frac{\pi }{{12}} + k\pi \\x = - \frac{{5\pi }}{{24}} + k\frac{\pi }{2}\end{array} \right.\)\(\left( {k \in \mathbb{Z}} \right)\)
d) Ta có \(\tan \frac{{2\pi }}{7} = \cot \left( {\frac{\pi }{2} - \frac{{2\pi }}{7}} \right) = \cot \frac{{3\pi }}{{14}}\).
Phương trình trở thành \(\cot 3x = \cot \frac{{3\pi }}{{14}} \Leftrightarrow 3x = \frac{{3\pi }}{{14}} + k\pi \Leftrightarrow x = \frac{\pi }{{14}} + k\frac{\pi }{3}\)\(\left( {k \in \mathbb{Z}} \right)\)
