Chủ đề này có 2 trả lời

[ductuMATHER]

Giải PT lượng giác sau:
$4sinx.sin(x+\frac{\Pi }{3}).sin(x+\frac{2\Pi }{3})+cos3x=1$
Thanks!

[conanthamtulungdanhkudo]

Giải PT lượng giác sau:
$4sinx.sin(x+\frac{\Pi }{3}).sin(x+\frac{2\Pi }{3})+cos3x=1$
Thanks!

$4sinx$$\frac{1}{2}(cos(-\pi /6)-cos(2x-\pi ))+cos3x=1$

$sinx-2sinxcos(2x+\pi )+cos3x=1\Leftrightarrow sinx+2sinxcos2x+cos3x=1$

$\Leftrightarrow sinx+2\frac{1}{2}\left [ sin(-x)+sin(3x) \right ]+cos3x=1$

$\Leftrightarrow sin3x+cos3x=1$

$\Leftrightarrow 2sin3xcos3x=0$

$\Leftrightarrow x=\frac{2k\pi }{3}$

$x=\frac{\pi }{6}+\frac{2k\pi }{3}$

[NTL2k1]

$4sinx$$\frac{1}{2}(cos(-\pi /6)-cos(2x-\pi ))+cos3x=1$

$sinx-2sinxcos(2x+\pi )+cos3x=1\Leftrightarrow sinx+2sinxcos2x+cos3x=1$

$\Leftrightarrow sinx+2\frac{1}{2}\left [ sin(-x)+sin(3x) \right ]+cos3x=1$

$\Leftrightarrow sin3x+cos3x=1$ (*)

$\Leftrightarrow 2sin3xcos3x=0$

$\Leftrightarrow x=\frac{2k\pi }{3}$

$x=\frac{\pi }{6}+\frac{2k\pi }{3}$

Có sai sót rồi bạn 

Chỗ màu đỏ phải là $x=\frac{\pi }{2}+\frac{k2\pi }{3}$

Cụ thể

(*) $\Leftrightarrow$ $\sqrt{2}sin(3x+\frac{\pi }{4})=1$

$\Leftrightarrow sin(3x+\frac{\pi }{4})=\frac{1}{\sqrt{2}}=sin\frac{\pi }{4}$

$\Leftrightarrow$ $\left[ {\begin{array}{*{20}{c}} {3x+\frac{\pi }{4}=\frac{\pi }{4}+k2\pi} \\ {3x+\frac{\pi }{4}=\frac{7\pi }{4}+k2\pi} \\ \end{array}} \right.$

$\Leftrightarrow$ $\left[ {\begin{array}{*{20}{c}} {x=\frac{k2\pi }{3}} \\ {x=\frac{\pi }{2}+\frac{k2\pi }{3}} \\ \end{array}} \right.$