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Консультация № 95840
20.07.2007, 23:40
0.00 руб.
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Помогите пожалуйста решить уравнение
arcsin x * arccos x = -1
Буду очень признательна за помощь, спасибо...
arcsin x * arccos x = -1
Буду очень признательна за помощь, спасибо...
Обсуждение
Неизвестный
21.07.2007, 00:51
общий
это ответ
Здравствуйте, Lan!
Напомним, что арксинус и арккосинус имеют следующие свойства (для любого -1<=x<=1):
1. 0<=arccos(x)<=pi (pi приблизительно равно 3.14159);
2. -pi/2<=arcsin(x)<=pi/2;
3. arcsin(x) + arccos(x) = pi/2.
Воспользуемся этими свойствами.
Так как арккосинус всегда неотрицателен (см. 1), значит, arcsin(x)*arccos(x)=-1 только если arcsin(x)<0. (*)
Введём переменную t=arcsin(x). Тогда из свойства 3 следует, что arccos(x)=pi/2-t. Составим и решим уравнение:
t * (pi/2-t) = -1 (умножим обе части на 2 и раскроем скобки)
pi*t - 2t^2 = -2 (перенесём все слагаемые в левую часть и умножим на -1)
2t^2 - pi*t -2 = 0
дискриминант D = (-pi)^2 - 4*2*(-2) = pi^2 + 16 > 0 - уравнение имеет два корня
t1 = (pi + sqrt(pi^2+16))/4 > 0 - не подходит (см. (*))
t2 = (pi - sqrt(pi^2+16))/4 < 0.
Перейдём обратно к переменной x:
arcsin(x) = (pi - sqrt(pi^2+16))/4
x = sin((pi - sqrt(pi^2+16))/4).
Ответ: x = sin((pi - sqrt(pi^2+16))/4).
Напомним, что арксинус и арккосинус имеют следующие свойства (для любого -1<=x<=1):
1. 0<=arccos(x)<=pi (pi приблизительно равно 3.14159);
2. -pi/2<=arcsin(x)<=pi/2;
3. arcsin(x) + arccos(x) = pi/2.
Воспользуемся этими свойствами.
Так как арккосинус всегда неотрицателен (см. 1), значит, arcsin(x)*arccos(x)=-1 только если arcsin(x)<0. (*)
Введём переменную t=arcsin(x). Тогда из свойства 3 следует, что arccos(x)=pi/2-t. Составим и решим уравнение:
t * (pi/2-t) = -1 (умножим обе части на 2 и раскроем скобки)
pi*t - 2t^2 = -2 (перенесём все слагаемые в левую часть и умножим на -1)
2t^2 - pi*t -2 = 0
дискриминант D = (-pi)^2 - 4*2*(-2) = pi^2 + 16 > 0 - уравнение имеет два корня
t1 = (pi + sqrt(pi^2+16))/4 > 0 - не подходит (см. (*))
t2 = (pi - sqrt(pi^2+16))/4 < 0.
Перейдём обратно к переменной x:
arcsin(x) = (pi - sqrt(pi^2+16))/4
x = sin((pi - sqrt(pi^2+16))/4).
Ответ: x = sin((pi - sqrt(pi^2+16))/4).
Форма ответа
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