Giải bài 4 trang 34 sách bài tập toán 11 - Chân trời sáng tạo tập 1

Đề bài

Giải các phương trình lượng giác sau:

a) $\cos \left( {2x - \frac{\pi }{3}} \right) + \sin x$ $= 0$;

b) ${\cos ^2}\left( {x + \frac{\pi }{4}} \right)$ $= \frac{{2 + \sqrt 3 }}{4}$;

c) $\cos \left( {3x + \frac{\pi }{6}} \right) + 2{\sin ^2}x$ $= 1$

Lời giải chi tiết

a) $\cos \left( {2x - \frac{\pi }{3}} \right) + \sin x$ $= 0$ $\Leftrightarrow \cos \left( {2x - \frac{\pi }{3}} \right)$ $= \sin \left( { - x} \right)$ $\Leftrightarrow \cos \left( {2x - \frac{\pi }{3}} \right)$ $= \cos \left( {\frac{\pi }{2} + x} \right)$

 $\Leftrightarrow \left[ \begin{array}{l}2x - \frac{\pi }{3} = \frac{\pi }{2} + x + k2\pi \\2x - \frac{\pi }{3} =  - \left( {\frac{\pi }{2} + x} \right) + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)$ $\Leftrightarrow \left[ \begin{array}{l}x = \frac{{5\pi }}{6} + k2\pi \\x = \frac{{ - \pi }}{{18}} + \frac{{k2\pi }}{3}\end{array} \right.\left( {k \in \mathbb{Z}} \right)$

b) ${\cos ^2}\left( {x + \frac{\pi }{4}} \right)$ $= \frac{{2 + \sqrt 3 }}{4}$ $\Leftrightarrow \frac{{1 + \cos 2\left( {x + \frac{\pi }{4}} \right)}}{2}$ $= \frac{{2 + \sqrt 3 }}{4}$ $\Leftrightarrow \cos \left( {2x + \frac{\pi }{2}} \right)$ $= \cos \frac{\pi }{6}$

 $\Leftrightarrow \left[ {\begin{array}{*{20}{l}}{2x + \frac{\pi }{2} = \frac{\pi }{6} + k2\pi }\\{2x + \frac{\pi }{2} =  - \frac{\pi }{6} + k2\pi }\end{array}} \right.(k \in \mathbb{Z}{\rm{)}} \Leftrightarrow \left[ {\begin{array}{*{20}{l}}{x = \frac{{ - \pi }}{6} + k\pi }\\{x = \frac{{ - \pi }}{3} + k\pi }\end{array}} \right.\left( {k \in \mathbb{Z}} \right)$

c) $\cos \left( {3x + \frac{\pi }{6}} \right) + 2{\sin ^2}x$ $= 1$ $\Leftrightarrow \cos \left( {2x + \frac{\pi }{6}} \right)$ $= 1 - 2{\sin ^2}x$ $\Leftrightarrow \cos \left( {3x + \frac{\pi }{6}} \right)$ $= \cos 2x$

 $\Leftrightarrow \left[ \begin{array}{l}3x + \frac{\pi }{6} = 2x + k2\pi \\3x + \frac{\pi }{6} =  - 2x + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right)$ $\Leftrightarrow \left[ \begin{array}{l}x = \frac{{ - \pi }}{6} + k2\pi \\x = \frac{{ - \pi }}{{30}} + \frac{{k2\pi }}{5}\end{array} \right.\left( {k \in \mathbb{Z}} \right)$