1.
\(2sin\left(x+\dfrac{\pi}{4}\right)+\sqrt{3}=0\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=-\dfrac{\sqrt{3}}{2}\)
\(\Leftrightarrow sin\left(x+\dfrac{\pi}{4}\right)=sin\left(-\dfrac{\pi}{3}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\dfrac{\pi}{4}=-\dfrac{\pi}{3}+k2\pi\\x+\dfrac{\pi}{4}=\dfrac{4\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{7\pi}{12}+k2\pi\\x=\dfrac{13\pi}{12}+k2\pi\end{matrix}\right.\)
2.
\(2cos\left(x-60^o\right)-1=0\)
\(\Leftrightarrow cos\left(x-60^o\right)=\dfrac{1}{2}\)
\(\Leftrightarrow cos\left(x-60^o\right)=cos60^o\)
\(\Leftrightarrow\left[{}\begin{matrix}x-60^o=60^o+k.360^o\\x-60^o=-60^o+k.360^o\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=120^o+k.360^o\\x=k.360^o\end{matrix}\right.\)
3.
\(-4sin2x+1=0\)
\(\Leftrightarrow sin2x=\dfrac{1}{4}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=arcsin\dfrac{1}{4}+k2\pi\\2x=\pi-arcsin\dfrac{1}{4}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{2}arcsin\dfrac{1}{4}+k\pi\\x=\dfrac{\pi}{2}-\dfrac{1}{2}arcsin\dfrac{1}{4}+k\pi\end{matrix}\right.\)
4.
ĐK: \(x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
\(\sqrt{3}tan2x-1=0\)
\(\Leftrightarrow tan2x=\dfrac{1}{\sqrt{3}}\)
\(\Leftrightarrow tan2x=tan\dfrac{\pi}{6}\)
\(\Leftrightarrow2x=\dfrac{\pi}{6}+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{12}+\dfrac{k\pi}{2}\left(tm\right)\)
5.
\(cos\left(x+\dfrac{\pi}{3}\right)+sin2x=0\)
\(\Leftrightarrow cos\left(x+\dfrac{\pi}{3}\right)+cos\left(\dfrac{\pi}{2}-2x\right)=0\)
\(\Leftrightarrow2cos\left(\dfrac{5\pi}{12}+\dfrac{x}{2}\right).cos\left(\dfrac{3x}{2}-\dfrac{\pi}{12}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\left(\dfrac{5\pi}{12}+\dfrac{x}{2}\right)=0\\cos\left(\dfrac{3x}{2}-\dfrac{\pi}{12}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{5\pi}{12}+\dfrac{x}{2}=\pm\dfrac{\pi}{2}+k\pi\\\dfrac{3x}{2}-\dfrac{\pi}{12}=\pm\dfrac{\pi}{2}+k\pi\end{matrix}\right.\)
\(\Leftrightarrow...\)
6.
\(sin\left(x-50^o\right)=cos2x\)
\(\Leftrightarrow cos\left(140^o-x\right)=cos2x\)
\(\Leftrightarrow\left[{}\begin{matrix}140^o-x=2x+k.360^o\\140^o-x=-2x+k.360^o\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{3}.140^o-k.120^o\\x=-140^o+k.360^o\end{matrix}\right.\)
7.
ĐK: \(x\ne-60^o+k.540^o\)
\(3cot\left(\dfrac{x}{3}+20^o\right)+\sqrt{3}=0\)
\(\Leftrightarrow cot\left(\dfrac{x}{3}+20^o\right)=-\dfrac{\sqrt{3}}{3}\)
\(\Leftrightarrow cot\left(\dfrac{x}{3}+20^o\right)=cot\left(-60^o\right)\)
\(\Leftrightarrow\dfrac{x}{3}+20^o=-60^o+k.180^o\)
\(\Leftrightarrow x=-240^o+k.540^o\)
8.
\(4sinx.cosx.cos2x=-1\)
\(\Leftrightarrow2sin2x.cos2x=-1\)
\(\Leftrightarrow sin4x=-1\)
\(\Leftrightarrow4x=\pi+k2\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
9.
\(2sin\left(\dfrac{x}{2}+45^o\right)+2=0\)
\(\Leftrightarrow sin\left(\dfrac{x}{2}+45^o\right)=-1\)
\(\Leftrightarrow\dfrac{x}{2}+45^o=180^o+k.360^o\)
\(\Leftrightarrow x=270^o+k.360^o\)
10.
\(sin2x+2cosx=0\)
\(\Leftrightarrow2sinx.cosx+2cosx=0\)
\(\Leftrightarrow2cosx\left(sinx+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sinx=-1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=\pi+k2\pi\end{matrix}\right.\)
11.
\(sin^2x+cos2x=1\)
\(\Leftrightarrow sin^2x+1-2sin^2x=1\)
\(\Leftrightarrow sin^2x=0\)
\(\Leftrightarrow sinx=0\)
\(\Leftrightarrow x=k\pi\)
12.
\(sin^4x+cos^4x=-\dfrac{1}{2}cos^22x+1\)
\(\Leftrightarrow1-2sin^2x.cos^2x=-\dfrac{1}{2}cos^22x+1\)
\(\Leftrightarrow\dfrac{1}{2}sin^22x=\dfrac{1}{2}cos^22x\)
\(\Leftrightarrow cos^22x-sin^22x=0\)
\(\Leftrightarrow cos4x=0\)
\(\Leftrightarrow4x=\dfrac{\pi}{2}+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{8}+\dfrac{k\pi}{4}\)
