1.1.
\(sin\dfrac{x}{2}=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x}{2}=-\dfrac{\pi}{4}+k2\pi\\\dfrac{x}{2}=\dfrac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\pi}{2}+k2\pi\\x=\dfrac{5\pi}{2}+k2\pi\end{matrix}\right.\)
1.5.
\(cos\left(4x+\dfrac{\pi}{4}\right)=\dfrac{1}{3}\)
\(\Leftrightarrow4x+\dfrac{\pi}{4}=\pm arccos\left(\dfrac{1}{3}\right)+k2\pi\)
\(\Leftrightarrow x=-\dfrac{\pi}{16}\pm\dfrac{1}{4}arccos\left(\dfrac{1}{3}\right)+\dfrac{k\pi}{2}\)
1.6.
\(cos2x+cos\left(3x+\dfrac{\pi}{3}\right)=0\)
\(\Leftrightarrow2cos\left(\dfrac{5x}{2}+\dfrac{\pi}{6}\right).cos\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\left(\dfrac{5x}{2}+\dfrac{\pi}{6}\right)=0\\cos\left(\dfrac{x}{2}+\dfrac{\pi}{6}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{5x}{2}+\dfrac{\pi}{6}=\dfrac{\pi}{2}+k\pi\\\dfrac{x}{2}+\dfrac{\pi}{6}=\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{2\pi}{15}+\dfrac{k2\pi}{5}\\x=\dfrac{2\pi}{3}+k2\pi\end{matrix}\right.\)
2.1.
\(cos3x=-\dfrac{\sqrt{3}}{2}\)
\(\Leftrightarrow3x=\pm\dfrac{5\pi}{6}+k2\pi\)
\(\Leftrightarrow x=\pm\dfrac{5\pi}{18}+\dfrac{k2\pi}{3}\)
Ta có:
\(-\pi\le\pm\dfrac{5\pi}{18}+\dfrac{k2\pi}{3}\le\pi\)
...
2.2.
ĐK: \(x\in\left(k2\pi;2\pi+k2\pi\right)\)
\(\dfrac{2cos2x-1}{\sqrt{sinx}}=0\)
\(\Leftrightarrow2cos2x-1=0\)
\(\Leftrightarrow cos2x=\dfrac{1}{2}\)
\(\Leftrightarrow2x=\pm\dfrac{\pi}{3}+k2\pi\)
\(\Leftrightarrow x=\pm\dfrac{\pi}{6}+k\pi\)
3.1.
\(2sin\left(3x+5\right)=m+4\)
\(\Leftrightarrow sin\left(3x+5\right)=\dfrac{m+4}{2}\)
Phương trình có nghiệm khi:
\(\dfrac{m+4}{2}\in\left[-1;1\right]\)
\(\Leftrightarrow m+4\in\left[-2;2\right]\)
\(\Leftrightarrow m\in\left[-6;-2\right]\)
3.2.
\(\left(m+1\right)sinx=m+3\)
\(\Leftrightarrow sinx=\dfrac{m+3}{m+1}\left(m\ne-1\right)\)
Phương trình có nghiệm khi \(-1\le\dfrac{m+3}{m+1}\le1\Leftrightarrow m\le-2\)
