Вопрос пользователя:
решить систему уравнений:
а) {cos x + cos y = 1; {x + y = 2П
б) {sin x = cos y; {2 cos квадрат y + sin x = 3
Илюха отвечает:
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.b)quadleft{begin{matrix}sin x =cos y2cos^2y+sin x=3end{matrix}right.Rightarrow left{begin{matrix}sin x=cos y2cos^2y+cos y-3=0end{matrix}right.2cos^2y+cos y-3=0cos y=t,tinleft [ -1,1 right ]2t^2+t-3=0D=1-4cdot2cdot(-3)=25t_1=1,quad t_2=-frac32cos y=1Rightarrow y=2pi nleft{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." title=" 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.b)quadleft{begin{matrix}sin x =cos y2cos^2y+sin x=3end{matrix}right.Rightarrow left{begin{matrix}sin x=cos y2cos^2y+cos y-3=0end{matrix}right.2cos^2y+cos y-3=0cos y=t,tinleft [ -1,1 right ]2t^2+t-3=0D=1-4cdot2cdot(-3)=25t_1=1,quad t_2=-frac32cos y=1Rightarrow y=2pi nleft{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." alt=" 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.b)quadleft{begin{matrix}sin x =cos y2cos^2y+sin x=3end{matrix}right.Rightarrow left{begin{matrix}sin x=cos y2cos^2y+cos y-3=0end{matrix}right.2cos^2y+cos y-3=0cos y=t,tinleft [ -1,1 right ]2t^2+t-3=0D=1-4cdot2cdot(-3)=25t_1=1,quad t_2=-frac32cos y=1Rightarrow y=2pi nleft{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." />