Question

А10 Решение тригонометрических уравнений

Answer 1

$2Cos^{2} x-Sin^{2}x=-Sinx\\\\2(1-Sin^{2}x)-Sin^{2} x+Sinx=0\\\\2-2Sin^{2}x-Sin^{2} x+Sinx=0\\\\-3Sin^{2}x+Sinx+2=0\\\\3Sin^{2}x-Sinx-2=0\\\\Sinx=m,-1 \leq m \leq 1\\\\3m^{2}-m-2=0\\\\D=(-1)^{2}-4*3*(-2)=1+24=25=5^{2} \\\\m_{1}=\frac{1+5}{6}=1\\\\m_{2}=\frac{1-5}{6}=-\frac{2}{3}\\\\1)Sinx=1\\\\x=\frac{\pi }{2}+2\pi n,n\in Z\\\\2)Sinx=-\frac{2}{3}\\\\x=(-1)^{n}arcSin(-\frac{2}{3})+\pi n,n\in Z\\\\x=(-1)^{n+1}arcSin\frac{2}{3}+\pi n,n\in Z$