Question

даю 50 балов !!!!!!!!
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Answer 1

$\displaystyle\bf\\1)\\\\2Cos\frac{x}{2} -1=0\\\\2Cos\frac{x}{2} =1\\\\Cos\frac{x}{2} =\frac{1}{2} \\\\\frac{x}{2} =\pm \ arcCos\frac{1}{2} +2\pi n,n\in Z\\\\\frac{x}{2} =\pm \ \frac{\pi }{3} +2\pi n,n\in Z\\\\\boxed{x=\pm \ \frac{2\pi }{3} +4\pi n,n\in Z}\\\\\\2)\\\\Sin\frac{2}{3} x=\frac{1}{2} \\\\\frac{2}{3} x=(-1)^{n} arcSin\frac{1}{2}+\pi n,n\in Z\\\\\frac{2}{3} x=(-1)^{n} \frac{\pi }{6} +\pi n,n\in Z\\\\\boxed{x=(-1)^{n} \frac{\pi }{4}+\frac{3\pi n}{2} ,n\in Z}$

$\displaystyle\bf\\3)\\\\3Sin\Big(x-\frac{\pi }{6} \Big)=4\\\\Sin\Big(x-\frac{\pi }{6} \Big)=1\frac{1}{3} \\\\\boxed{x\in \oslash \ \ , \ \ tak \ kak \ -1\leq Sinx\leq 1}\\\\\\4)\\\\2Cos\Big(x-\frac{\pi }{3}\Big)=-2\\\\Cos\Big(x-\frac{\pi }{3} \Big)=-1\\\\x-\frac{\pi }{3} =\pi +2\pi n,n\in Z\\\\\boxed{x=\frac{4\pi }{3} +2\pi n,n\in Z}\\\\\\5)\\\\Cos2x Sin3x=Sin2x Cos3x\\\\Cos2x Sin3x-Sin2x Cos3x=0\\\\Sin(3x+2x)=0\\\\Sin5x=0\\\\5x=\pi n,n\in Z\\\\x=\frac{\pi n}{5} ,n\in Z$