Question

решить систему уравнений:

а) {cos x + cos y = 1; {x + y = 2П

б) {sin x = cos y; {2 cos квадрат y + sin x = 3

Answer 1

$<var>\ a)quadleft{begin{matrix} cos x+cos y=1\ x+y=2pi end{matrix}right.Rightarrow left{begin{matrix} cosleft ( 2pi-y right )+cos y=1\ x=2pi-y end{matrix}right.Rightarrow left{begin{matrix} cos y+cos y=1\ x=2pi-y end{matrix}right.Rightarrow\ Rightarrow left{begin{matrix} 2cos y=1\ x=2pi-y end{matrix}right.Rightarrow left{begin{matrix} cos y=frac12\ x=2pi-y end{matrix}right.Rightarrow left{begin{matrix} y=frac{pi}3+2pi n\ x=frac{5pi}{3}+2pi n end{matrix}right.\</var>b)quadleft{begin{matrix}sin x =cos y\2cos^2y+sin x=3end{matrix}right.Rightarrow left{begin{matrix}sin x=cos y\2cos^2y+cos y-3=0end{matrix}right.\2cos^2y+cos y-3=0\cos y=t,tinleft [ -1,1 right ]\2t^2+t-3=0\D=1-4cdot2cdot(-3)=25\t_1=1,quad t_2=-frac32\cos y=1Rightarrow y=2pi n\left{begin{matrix}sin x=1\ y=2pi nend{matrix}right.Rightarrow left{begin{matrix}x=frac{pi}{2}+2pi n\ y=2pi nend{matrix}right.$