Question

Очень нудна помощь в решении

Answer 1

Решить систему тригонометрических уравнений.

$displaystyle left { {{cos xcos y=sin^2y} atop {sin xsin y=cos^2y}} right. ~~~Big|~+\\ left { {{cos xcos y+sin xsin y=sin^2y+cos^2y} atop {sin xsin y=cos^2y}} right.\\\ left { {{cos xcos y+sin xsin y=1} atop {sin xsin y=cos^2y}} right.~~~~ left { {{cos (x- y)=1} atop {sin xsin y=cos^2y}} right.\\\ left { {{x- y=2pi n} atop {sin xsin y=cos^2y}} right.~~~~ left { {{x= y+2pi n} atop {sin (y+ 2pi n)sin y=cos^2y}} right.$

$displaystyle left { {{x= y+2pi n} atop {sin y sin y=1-sin^2y}} right.~~~~ left { {{x= y+2pi n} atop {2sin^2 y=1}} right.\\\ left { {{x= y+2pi n} atop {sin^2 y=dfrac 12}} right.~~~~ left { {{x= y+2pi n} atop {sin y=pm dfrac 1{sqrt2}}} right.~~~Leftrightarrow~~~ left { {{x= y+2pi n} atop {sin y=pm dfrac {sqrt2}2}} right.$

$boxed {y=dfrac {pi}4+dfrac{pi}2 k,~~~x=dfrac {pi}4+dfrac{pi}2 k+2pi n;~~~k,nin Z }$

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Использованы формулы

$sin^2 y + cos^2 y = 1\ cos x cos y + sin x sin y = cos (x-y)$