a: \(sin5x=-\dfrac{1}{2}\)
=>\(\left[{}\begin{matrix}5x=-\dfrac{\Omega}{6}+k2\Omega\\5x=\dfrac{7}{6}\Omega+k2\Omega\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-\dfrac{\Omega}{30}+\dfrac{k2\Omega}{5}\\x=\dfrac{7}{30}\Omega+\dfrac{k2\Omega}{5}\end{matrix}\right.\)
b: ĐKXĐ: \(x-\dfrac{\Omega}{12}\ne k\Omega\)
=>\(x=\dfrac{\Omega}{12}+k\Omega\)
\(\sqrt{3}\cdot cot\left(x-\dfrac{\Omega}{12}\right)=1\)
=>\(cot\left(x-\dfrac{\Omega}{12}\right)=\dfrac{1}{\sqrt{3}}\)
=>\(x-\dfrac{\Omega}{12}=\dfrac{\Omega}{3}+k\Omega\)
=>\(x=\dfrac{\Omega}{3}+\dfrac{\Omega}{12}+k\Omega=\dfrac{5}{12}\Omega+k\Omega\)
c:
cos3x-sinx=0
=>\(cos3x=sinx\)
=>\(cos3x=cos\left(\dfrac{\Omega}{2}-x\right)\)
=>\(\left[{}\begin{matrix}3x=\dfrac{\Omega}{2}-x+k2\Omega\\3x=x-\dfrac{\Omega}{2}+k2\Omega\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}4x=\dfrac{\Omega}{2}+k2\Omega\\2x=-\dfrac{\Omega}{2}+k2\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=\dfrac{\Omega}{4}+k\Omega\\x=-\dfrac{\Omega}{4}+k\Omega\end{matrix}\right.\)
d:
ĐKXĐ: \(x\ne k\Omega;x\ne\dfrac{\Omega}{4}+\dfrac{k\Omega}{2}\)
\(tan2x+cotx=0\)
=>\(tan2x=-cotx=cot\left(-x\right)\)
=>\(tan2x=tan\left(\dfrac{\Omega}{2}+x\right)\)
=>\(2x=x+\dfrac{\Omega}{2}+k\Omega\)
=>\(x=\dfrac{\Omega}{2}+k\Omega\)
Kết hợp ĐKXĐ, ta được: \(x=\dfrac{\Omega}{2}+k\Omega\)
e: ĐKXĐ: \(x\ne\dfrac{\Omega}{2}+k\Omega;x\ne k\Omega\)
\(tanx+cotx=2\)
=>\(tanx+\dfrac{1}{tanx}=2\)
=>\(\dfrac{tan^2x+1}{tanx}=2\)
=>\(tan^2x+1-2tanx=0\)
=>\(\left(tanx-1\right)^2=0\)
=>tan x-1=0
=>tan x=1
=>\(x=\dfrac{\Omega}{4}+k\Omega\)
