Giải phương trình cos x + cos 3 x + 2 cos 5 x = 0

* Hướng dẫn giải

\[\cos x + \cos 3x + 2\cos 5x = 0\]

\[ \Leftrightarrow cosx + cos3x + cos5x + cos5x = 0\]

\[ \Leftrightarrow (cosx + cos5x) + (cos3x + cos5x) = 0\]

\[ \Leftrightarrow 2cos3xcos2x + 2cos4xcosx = 0\]

\[ \Leftrightarrow 2(4co{s^3}x - 3cosx)cos2x + 2cos4xcosx = 0\]

\[ \Leftrightarrow 2cosx(4co{s^2}x - 3)cos2x + 2cos4xcosx = 0\]

\[ \Leftrightarrow 2cosx[(4co{s^2}x - 3)cos2x + cos4x] = 0\]

\[ \Leftrightarrow 2cosx[[2(1 + cos2x) - 3]cos2x + 2co{s^2}2x - 1] = 0\]

\[ \Leftrightarrow 2cosx[(2cos2x - 1)cos2x + 2co{s^2}2x - 1] = 0\]

\[ \Leftrightarrow 2cosx[4co{s^2}2x - cos2x - 1] = 0\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{cosx = 0}\\{cos2x = \frac{{1 + \sqrt {17} }}{8}}\\{cos2x = \frac{{1 - \sqrt {17} }}{8}}\end{array}} \right.\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{2} + k\pi }\\{2x = \pm arccos\frac{{1 + \sqrt {17} }}{8} + k2\pi }\\{2x = \pm arccos\frac{{1 - \sqrt {17} }}{8} + k2\pi }\end{array}} \right.\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = \frac{\pi }{2} + k\pi }\\{x = \pm \frac{1}{2}arccos\frac{{1 + \sqrt {17} }}{8} + k\pi }\\{x = \pm \frac{1}{2}arccos\frac{{1 - \sqrt {17} }}{8} + k\pi }\end{array}} \right.\)

Vậy nghiệm của phương trình là: \[x = \frac{\pi }{2} + k\pi ,x = \pm \frac{1}{2}\arccos \frac{{1 + \sqrt {17} }}{8} + k\pi ,x = \pm \frac{1}{2}\arccos \frac{{1 - \sqrt {17} }}{8} + k\pi \]

Đáp án cần chọn là: DCâu 32. Giải phương trình \[\sin 3x - \sin x + \sin 2x = 0\]

A.\[x = k\pi ,x = \frac{\pi }{3} + \frac{{k2\pi }}{3}\]

B. \[x = \pm \frac{\pi }{3} + \frac{{k2\pi }}{3}\]

C. \[x = \frac{\pi }{2} + k\pi ,x = - \frac{\pi }{3} + \frac{{k2\pi }}{3}\]

D. \[x = 2k\pi ,x = \frac{\pi }{2} + \frac{{k\pi }}{3}\]Trả lời:

\[\sin 3x - \sin x + \sin 2x = 0\]

\[ \Leftrightarrow 2cos2xsinx + 2sinxcosx = 0\]

\[ \Leftrightarrow 2sinx(cos2x + cosx) = 0\]

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{sinx = 0}\\{cos2x = - cosx = cos(\pi - x)}\end{array}} \right.\)

\( \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = k\pi }\\{\begin{array}{*{20}{c}}{2x = \pi - x + k2\pi }\\{2x = x - \pi + k2\pi }\end{array}}\end{array}} \right.\)

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

Vậy nghiệm của phương trình là:\[x = k\pi ,x = \frac{\pi }{3} + \frac{{k2\pi }}{3}\]

Đáp án cần chọn là: A