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Консультация № 161915
03.03.2009, 16:05
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Здравствуйте ! помогите решить уравнение 2arctg(x)=x .
Обсуждение
03.03.2009, 20:47
общий
это ответ
Здравствуйте, Matemateg!
Учитывая область допустимых значений функции арктангенс, получим -π/2 < x < π/2.
Простейший способ решения заданного уравнения – графический. Он заключается в нахождении абсцисс точек пересечения графиков функций y = arctg x и y = x/2…
Можно видоизменить задачу. Применим к обеим частям уравнения функцию tg x:
tg (2arctg x) = tg x.
Обозначим α = arctg x. Тогда
tg (2arctg x) = tg 2α = (2tg α)/(1 – (tg α)^2) = 2x/(1 – x^2),
2x/(1 – x^2) = tg x.
Полученное уравнение (как и исходное) неразрешимо в радикалах. Для его решения следует использовать численные методы. Можно, к тому же, приближенно решить его с помощью разложения функции тангенс в степенной ряд, взяв соответствующее требуемой точности решения число членов ряда.
Не исключено, что я ошибаюсь...
С уважением.
Учитывая область допустимых значений функции арктангенс, получим -π/2 < x < π/2.
Простейший способ решения заданного уравнения – графический. Он заключается в нахождении абсцисс точек пересечения графиков функций y = arctg x и y = x/2…
Можно видоизменить задачу. Применим к обеим частям уравнения функцию tg x:
tg (2arctg x) = tg x.
Обозначим α = arctg x. Тогда
tg (2arctg x) = tg 2α = (2tg α)/(1 – (tg α)^2) = 2x/(1 – x^2),
2x/(1 – x^2) = tg x.
Полученное уравнение (как и исходное) неразрешимо в радикалах. Для его решения следует использовать численные методы. Можно, к тому же, приближенно решить его с помощью разложения функции тангенс в степенной ряд, взяв соответствующее требуемой точности решения число членов ряда.
Не исключено, что я ошибаюсь...
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
04.03.2009, 06:58
общий
В теории замечательных кривых (например, циклоида) встречаются уравнения типа
x = tg x
x + Pi = tg x
Их решением являются некоторые известные иррациональные числа, я так понимаю, невыразимые известными нам корнями, арктангенсами и т.д.
x = tg x
x + Pi = tg x
Их решением являются некоторые известные иррациональные числа, я так понимаю, невыразимые известными нам корнями, арктангенсами и т.д.
Неизвестный
12.03.2009, 21:40
общий
по-моему если рассматривать диапазон аргумента тангенса от -pi/2 до pi/2.
(При этом кстати не "-π/2 < x < π/2", а "-pi/2 < 2*x < pi/2", то есть "-pi/4 < x < pi/4")
то решение уравнения имеет один корень
2*arctg(x)=x
arctg(x)= 2*x
x= tg(2*x)
x= 0
А если рассматривать весь диапазон, то по-моему, там уже конечно бесконечное количество иррациональных.
(При этом кстати не "-π/2 < x < π/2", а "-pi/2 < 2*x < pi/2", то есть "-pi/4 < x < pi/4")
то решение уравнения имеет один корень
2*arctg(x)=x
arctg(x)= 2*x
x= tg(2*x)
x= 0
А если рассматривать весь диапазон, то по-моему, там уже конечно бесконечное количество иррациональных.
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