Question

30 БАЛЛОВ решите, пожалуйста, уравнения

Answer 1

$1)2Cos^{2}3x=3Sin3x\\\\2(1-Sin^{2}3x)-3Sin3x=0\\\\2-2Sin^{2}3x-3Sin3x=0\\\\2Sin^{2}3x+3Sin3x-2=0\\\\Sin3x=m, \ -1\leq m\leq1\\\\2m^{2}+3m-2=0\\\\D=3^{2}-4*2*(-2)=9+16=25=5^{2}\\\\m_{1}=\frac{-3+5}{4}=\frac{1}{2}\\\\m_{2}=\frac{-3-5}{4}=-2<-1-neyd\\\\Sin3x=\frac{1}{2}\\\\3x=(-1)^{n}arcSin\frac{1}{2}+\pi n,n\in Z\\\\3x=(-1)^{n}\frac{\pi }{6}+\pi n,n\in Z\\\\\boxed{x=(-1)^{n}\frac{\pi }{18}+\frac{\pi n }{3},n\in Z}$

$2)7Sin^{2}x-4Sin2x+Cos^{2}x=0\\\\7Sin^{2}x-8Sinx Cosx+Cos^{2}x=0 \ |:Cos^{2} x \ ,Cosx \neq 0\\\\7tg^{2}x-8tgx+1=0\\\\tgx=m\\\\7m^{2}-8m+1=0\\\\D=(-8)^{2} -4*7*1=64- 28=36=6^{2}\\\\m_{1}=\frac{8-6}{14}=\frac{1}{7}\\\\m_{2}=\frac{8+6}{14}=1\\\\1)tgx=\frac{1}{7} \\\\\boxed{x'=arctg\frac{1}{7} +\pi n,n\in Z}\\\\2)tgx=1\\\\x=arctg1+\pi n,n\in Z\\\\\boxed{x''=\frac{\pi }{4}+\pi n,n\in Z}$