42: \(cot\left(x-\dfrac{\Omega}{5}\right)=-\sqrt{3}\)
=>\(tan\left(x-\dfrac{\Omega}{5}\right)=-\dfrac{1}{\sqrt{3}}=tan\left(-\dfrac{\Omega}{6}\right)\)
=>\(x-\dfrac{\Omega}{5}=-\dfrac{\Omega}{6}+k\Omega\)
=>\(x=-\dfrac{\Omega}{6}+\dfrac{\Omega}{5}+k\Omega=\dfrac{\Omega}{30}+k\Omega\)
43: \(sin\left(\Omega-x\right)-cosx=0\)
=>\(sin\left(\Omega-x\right)=cosx=sin\left(\dfrac{\Omega}{2}-x\right)\)
=>\(sin\left(x-\Omega\right)=sin\left(x-\dfrac{\Omega}{2}\right)\)
=>\(\left[{}\begin{matrix}x-\Omega=x-\dfrac{\Omega}{2}+k2\Omega\\x-\Omega=\Omega-x+\dfrac{\Omega}{2}+k2\Omega\end{matrix}\right.\)
=>\(x-\Omega=\dfrac{3}{2}\Omega-x+k2\Omega\)
=>\(2x=\dfrac{5}{2}\Omega+k2\Omega\)
=>\(x=\dfrac{5}{4}\Omega+k\Omega\)
45: \(sinx\cdot sin\left(x-\dfrac{\Omega}{18}\right)=0\)
=>\(\left[{}\begin{matrix}sinx=0\\sin\left(x-\dfrac{\Omega}{18}\right)=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=k\Omega\\x-\dfrac{\Omega}{18}=k\Omega\end{matrix}\right.\)
=>\(\left[{}\begin{matrix}x=k\Omega\\x=\dfrac{\Omega}{18}+k\Omega\end{matrix}\right.\)
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