1.
\(sin\left(2x-1\right)=sin\left(3x+1\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x-1=3x+1+k2\pi\\2x-1=\pi-3x-1+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-2-k2\pi\\x=\dfrac{\pi}{5}+\dfrac{k2\pi}{5}\end{matrix}\right.\)
Vậy phương trình đã cho có nghiệm \(x=-2-k2\pi;x=\dfrac{\pi}{5}+\dfrac{k2\pi}{5}\)
2.
\(cos\left(x-\dfrac{\pi}{4}\right)=cos\left(2x+\dfrac{\pi}{2}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{4}=2x+\dfrac{\pi}{2}+k2\pi\\x-\dfrac{\pi}{4}=-2x-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{3\pi}{4}-k2\pi\\x=-\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)
Vậy phương trình đã cho có nghiệm \(x=-\dfrac{3\pi}{4}-k2\pi;x=-\dfrac{\pi}{4}+k2\pi\)
3.
ĐK: \(x\ne-\dfrac{3}{2}+\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
\(tan\left(2x+3\right)=tan\dfrac{\pi}{3}\)
\(\Leftrightarrow2x+3=\dfrac{\pi}{3}+k\pi\)
\(\Leftrightarrow x=-\dfrac{3}{2}+\dfrac{\pi}{6}+\dfrac{k\pi}{2}\)
Vậy phương trình đã cho có nghiệm \(x=-\dfrac{3}{2}+\dfrac{\pi}{6}+\dfrac{k\pi}{2}\)
4.
\(cot\left(45^o-x\right)=\dfrac{\sqrt{3}}{3}\)
\(\Leftrightarrow cot\left(45^o-x\right)=cot60^o\)
\(\Leftrightarrow45^o-x=60^o+k180^o\)
\(\Leftrightarrow x=-15^o-k180^o\)
Vậy phương trình đã cho có nghiệm \(x=-15^o-k180^o\)
5.
\(sin2x=\dfrac{\sqrt{3}}{2}\)
\(\Leftrightarrow sin2x=sin\dfrac{\pi}{3}\)
\(\Leftrightarrow\left[{}\begin{matrix}2x=\dfrac{\pi}{3}+k2\pi\\2x=\pi-\dfrac{\pi}{3}+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{6}+k\pi\\x=\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
Vậy phương trình đã cho có nghiệm \(x=\dfrac{\pi}{6}+k\pi;x=\dfrac{\pi}{3}+k\pi\)
6.
\(cos\left(2x+25^o\right)=-\dfrac{\sqrt{2}}{2}\)
\(\Leftrightarrow cos\left(2x+25^o\right)=cos135^o\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+25^o=135^o+k360^o\\2x+25^o=-135^o+k360^o\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=55^o+k180^o\\x=-80^o+k180^o\end{matrix}\right.\)
Vậy phương trình đã cho có nghiệm \(x=55^o+k180^o;x=-80^o+k180^o\)
