Question

Помогите Я на уроке не мог решить хочу узнать как это решается
система уравнений
sinx+cosy=1/2
siny-cosx=1/2

Answer 1

$left { {{sin(x)+cos(y)=frac{1}{2}} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{sin(x)+cos(y)-[sin(y)-cos(x)]=frac{1}{2}-frac{1}{2}} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{[sin(x)+cos(x)]-[sin(y)-cos(y)]=0} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{sqrt{2}sin(x+frac{pi}{4})-sqrt{2}sin(y-frac{pi}{4})=0} atop {sin(y)-cos(x)=frac{1}{2}}} right.$

$left { {{sin(x+frac{pi}{4})-sin(y-frac{pi}{4})=0} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{2sin(frac{x-y}{2}+frac{pi}{4})*cos(frac{x+y}{2})=0} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{frac{x-y}{2}+frac{pi}{4}=pi n, nin Z or frac{x+y}{2}=frac{pi}{2}+pi n, nin Z} atop {sin(y)-cos(x)=frac{1}{2}}} right.$

$left { {{x-y=-frac{pi}{2}+2pi n, nin Z or x+y=pi+2pi n, nin Z} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{y=x+frac{pi}{2}-2pi n, nin Z or y=-x+pi+2pi n, nin Z} atop {sin(y)-cos(x)=frac{1}{2}}} right. \\ left { {{y=x+frac{pi}{2}-2pi n, nin Z} atop {sin[x+frac{pi}{2}-2pi n]-cos(x)=frac{1}{2}}} right. or left { {{y=-x+pi+2pi n, nin Z} atop {sin(-x+pi+2pi n)-cos(x)=frac{1}{2}}} right.$

$left { {{y=x+frac{pi}{2}-2pi n, nin Z} atop {sin(frac{pi}{2}+x)-cos(x)=frac{1}{2}}} right. or left { {{y=-x+pi+2pi n, nin Z} atop {sin(pi-x)-cos(x)=frac{1}{2}}} right. \\ left { {{y=x+frac{pi}{2}-2pi n, nin Z} atop {cos(x)-cos(x)=frac{1}{2}}} right. or left { {{y=-x+pi+2pi n, nin Z} atop {sin(x)-cos(x)=frac{1}{2}}} right. \\ left { {{y=-x+pi+2pi n, nin Z} atop {sin(x)-cos(x)=frac{1}{2}}} right.$

$left { {{y=-x+pi+2pi n, nin Z} atop {sqrt{2}sin(x-frac{pi}{4})=frac{1}{2}}} right. \\ left { {{y=-x+pi+2pi n, nin Z} atop {sin(x-frac{pi}{4})=frac{1}{2sqrt{2}}}} right. \\ left { {{y=-x+pi+2pi n, nin Z} atop {x-frac{pi}{4}=(-1)^k*arcsin(frac{1}{2sqrt{2}})+pi k, kin Z}} right. \\ left { {{y=-x+pi+2pi n, nin Z} atop {x=frac{pi}{4}+(-1)^k*arcsin(frac{1}{2sqrt{2}})+pi k, kin Z}} right.$

$left { {{y=-frac{pi}{4}-(-1)^k*arcsin(frac{1}{2sqrt{2}})-pi k+pi+2pi n, nin Z, kin Z} atop {x=frac{pi}{4}+(-1)^k*arcsin(frac{1}{2sqrt{2}})+pi k, kin Z}} right. \\ left { {{y=-frac{pi}{4}+(-1)^{k+1}*arcsin(frac{1}{2sqrt{2}})+pi k+2pi n, nin Z, kin Z} atop {x=frac{pi}{4}+(-1)^k*arcsin(frac{1}{2sqrt{2}})+pi k, kin Z}} right.$