1.
\(\Leftrightarrow2sinx.cosx=0\)
\(\Leftrightarrow sin2x=0\)
\(\Leftrightarrow2x=k\pi\)
\(\Leftrightarrow x=\frac{k\pi}{2}\)
2.
\(\Leftrightarrow\frac{\sqrt{2}}{2}sinx+\frac{\sqrt{2}}{2}cosx=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\left(x+\frac{\pi}{4}\right)=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\frac{\pi}{4}=-\frac{\pi}{4}+k2\pi\\x+\frac{\pi}{4}=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{2}+k2\pi\\x=\pi+k2\pi\end{matrix}\right.\)
3.
\(\Leftrightarrow2.\left(2sinx.cosx\right)cos2x=0\)
\(\Leftrightarrow2sin2x.cos2x=0\)
\(\Leftrightarrow sin4x=0\)
\(\Leftrightarrow4x=k\pi\)
\(\Leftrightarrow x=\frac{k\pi}{4}\)
4.
\(\Leftrightarrow\left(sinx-1\right)\left(sinx-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}sinx=1\\sinx=3>1\left(l\right)\end{matrix}\right.\)
\(\Leftrightarrow x=\frac{\pi}{2}+k2\pi\)
5.
\(\Leftrightarrow\frac{\sqrt{2}}{2}sin2x+\frac{\sqrt{2}}{2}cos2x=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin2x.sin\frac{\pi}{4}+cos2x.cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow sin\left(2x+\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+\frac{\pi}{4}=\frac{\pi}{4}+k2\pi\\2x+\frac{\pi}{4}=\frac{3\pi}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\frac{\pi}{4}+k\pi\end{matrix}\right.\)
6.
\(\Leftrightarrow2sin2x=-1\)
\(\Leftrightarrow sin2x=-\frac{1}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=-\frac{\pi}{6}+k2\pi\\2x=\frac{7\pi}{6}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{12}+k\pi\\x=\frac{7\pi}{12}+k\pi\end{matrix}\right.\)
7.
\(\Leftrightarrow\left[{}\begin{matrix}2x-40^0=60^0+k360^0\\2x-40^0=120^0+n360^0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=50^0+k180^0\\x=80^0+n180^0\end{matrix}\right.\)
Do \(-180^0\le x\le180^0\Rightarrow\left\{{}\begin{matrix}-180^0\le50^0+k180^0\le180^0\\-180^0\le80^0+n180^0\le180^0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}-\frac{23}{18}\le k\le\frac{13}{18}\\-\frac{13}{9}\le n\le\frac{5}{9}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}k=\left\{-1;0\right\}\\n=\left\{-1;0\right\}\end{matrix}\right.\)
\(\Rightarrow x=\left\{-130^0;50^0;-100^0;80^0\right\}\)
8.
\(\Leftrightarrow sinx=-\frac{\sqrt{2}}{2}\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{4}+k2\pi\\x=\frac{5\pi}{4}+k2\pi\end{matrix}\right.\)
