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Консультация № 194698
13.02.2019, 19:47
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе:
Найти наибольшее и наименьшее значение функции
y=a×(cos^2(x))+2×b×(sin(x)) ×(cos(x)) +c×(sin^2(x))
Найти наибольшее и наименьшее значение функции
y=a×(cos^2(x))+2×b×(sin(x)) ×(cos(x)) +c×(sin^2(x))
Обсуждение
?
13.02.2019, 20:45
общий
это ответ
Здравствуйте, alina.poroshina124564!
Выражаем через косинус удвоенного аргумента
cos2(x)=0.5+0.5cos(2x)
sin2(x)=0.5-0.5cos(2x)
sin(x)[$183$]cos(x)=0.5sin(2x)=0.5cos(2x-[$960$]/2)
исходная функция таким образом раскладывается на непериодическую и периодическую составляющие
y=a[$183$]cos2(x)+2b[$183$]sin(x)[$183$]cos(x)+c[$183$]sin2(x)=0.5a+0.5c+(0.5a-0.5c)cos(2x)+b[$183$]cos(2x-[$960$]/2)=0.5a+0.5c+r[$183$]cos(2x+[$966$]0)
чтобы сложить периодические составляющие, достаточно представить их аргументы в виде векторов на комплексной плоскости в при некотором значении x (например, x=0) и сложить эти векторы
r=[$8730$]((0.5a-0.5c)2+b2)
[$966$]0=-arcctg((0.5a-0.5c)/b) (впрочем, это значение нам не понадобится)
y=a[$183$]cos2(x)+2b[$183$]sin(x)[$183$]cos(x)+c[$183$]sin2(x)=0.5a+0.5c+[$8730$]((0.5a-0.5c)2+b2)[$183$]cos(2x+[$966$]0)
ymin=0.5a+0.5c-[$8730$]((0.5a-0.5c)2+b2)
ymax=0.5a+0.5c+[$8730$]((0.5a-0.5c)2+b2)
Выражаем через косинус удвоенного аргумента
cos2(x)=0.5+0.5cos(2x)
sin2(x)=0.5-0.5cos(2x)
sin(x)[$183$]cos(x)=0.5sin(2x)=0.5cos(2x-[$960$]/2)
исходная функция таким образом раскладывается на непериодическую и периодическую составляющие
y=a[$183$]cos2(x)+2b[$183$]sin(x)[$183$]cos(x)+c[$183$]sin2(x)=0.5a+0.5c+(0.5a-0.5c)cos(2x)+b[$183$]cos(2x-[$960$]/2)=0.5a+0.5c+r[$183$]cos(2x+[$966$]0)
чтобы сложить периодические составляющие, достаточно представить их аргументы в виде векторов на комплексной плоскости в при некотором значении x (например, x=0) и сложить эти векторы
r=[$8730$]((0.5a-0.5c)2+b2)
[$966$]0=-arcctg((0.5a-0.5c)/b) (впрочем, это значение нам не понадобится)
y=a[$183$]cos2(x)+2b[$183$]sin(x)[$183$]cos(x)+c[$183$]sin2(x)=0.5a+0.5c+[$8730$]((0.5a-0.5c)2+b2)[$183$]cos(2x+[$966$]0)
ymin=0.5a+0.5c-[$8730$]((0.5a-0.5c)2+b2)
ymax=0.5a+0.5c+[$8730$]((0.5a-0.5c)2+b2)
5
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