g, \(1+2sinx=2cosx\)
\(\Leftrightarrow sinx-cosx=-\dfrac{1}{2}\)
\(\Leftrightarrow\sqrt{2}sin\left(x-\dfrac{\pi}{4}\right)=-\dfrac{1}{2}\)
\(\Leftrightarrow sin\left(x-\dfrac{\pi}{4}\right)=-\dfrac{1}{2\sqrt{2}}\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{4}=arcsin\left(-\dfrac{1}{2\sqrt{2}}\right)+k2\pi\\x-\dfrac{\pi}{4}=\pi-arcsin\left(-\dfrac{1}{2\sqrt{2}}\right)+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{4}+arcsin\left(-\dfrac{1}{2\sqrt{2}}\right)+k2\pi\\x=\dfrac{5\pi}{4}-arcsin\left(-\dfrac{1}{2\sqrt{2}}\right)+k2\pi\end{matrix}\right.\)
h, \(4cosx-3sinx=3\)
\(\Leftrightarrow5\left(\dfrac{4}{5}cosx-\dfrac{3}{5}sinx\right)=3\)
\(\Leftrightarrow cos\left(x+arccos\dfrac{4}{5}\right)=\dfrac{3}{5}\)
\(\Leftrightarrow x+arccos\dfrac{4}{5}=\pm arccos\dfrac{3}{5}+k2\pi\)
\(\Leftrightarrow x=-arccos\dfrac{4}{5}\pm arccos\dfrac{3}{5}+k2\pi\)
i, \(3cos3x+4sin3x=5\)
\(\Leftrightarrow\dfrac{3}{5}cos3x+\dfrac{4}{5}sin3x=1\)
\(\Leftrightarrow cos\left(3x-arccos\dfrac{3}{5}\right)=1\)
\(\Leftrightarrow3x-arccos\dfrac{3}{5}=k2\pi\)
\(\Leftrightarrow x=\dfrac{1}{3}arccos\dfrac{3}{5}+\dfrac{k2\pi}{3}\)
