Question

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Answer 1

$2cos^2(3x)+sin(5x)=1 medskip \ left(2cos^2(3x)-1right)+sin(5x)=0 medskip \ cos(6x)+sin(5x)=0 medskip \ sinleft(dfrac{pi}{2}-6xright)+sin(5x)=0 medskip \ 2sinleft(dfrac{frac{pi}{2}-6x+5x}{2}right)cosleft(dfrac{frac{pi}{2}-6x-5x}{2}right)=0 medskip \ sinleft(dfrac{x}{2}-dfrac{pi}{4}right)cosleft(dfrac{11x}{2}-dfrac{pi}{4}right)=0$

$1)~sinleft(dfrac{x}{2}-dfrac{pi}{4}right)=0 medskip \ dfrac{x}{2}-dfrac{pi}{4}=pi m,~minmathbb{Z} medskip \ x=dfrac{pi}{2}+2pi m,~minmathbb{Z} medskip \ 2)~cosleft(dfrac{11x}{2}-dfrac{pi}{4}right)=0 medskip \ dfrac{11x}{2}-dfrac{pi}{4}=dfrac{pi}{2}+pi k,~kinmathbb{Z} medskip \ x=dfrac{3pi}{22}+dfrac{2pi k}{11},~kinmathbb{Z}$

Ответ. $x=dfrac{pi}{2}+2pi m,~minmathbb{Z};~x=dfrac{3pi}{22}+dfrac{2pi k}{11},~kinmathbb{Z}$