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Задать вопрос экспертам

Консультация № 176505

05.02.2010, 04:31

0.00 руб.

0 1 1

Образование

Уважаемые эксперты, помогите пожалуйста решить задание.
Решить уравнение cosx=(asinx-1)^0,5 при всех значениях параметра а.

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

05.02.2010, 11:17

общий

это ответ

Здравствуйте, STASSY.

Уравнение равносильно системе
cos²x=asin x-1
cos x[$8805$]0

Решаем первое уравнение, заменяя cos²x на 1-sin²x, получаем
sin²x+asin x-2=0
sin x=0.5[-a[$177$][$8730$](a²+8)]

Решая неравенство -1[$8804$]0.5[-a+[$8730$](a²+8)][$8804$]1, получаем a[$8805$]1
Решая неравенство -1[$8804$]0.5[-a-[$8730$](a²+8)][$8804$]1, получаем a[$8804$]-1

1) a[$8805$]1, подходит только корень со знаком плюс, учитывая что cos x[$8805$]0, получаем решение
x=arcsin(0.5[-a+[$8730$](a²+8)])+2 pi n (n - целое)

2) -1<a<1 решений нет

3) a[$8804$]-1, подходит только корень со знаком минус, учитывая что cos x[$8805$]0, получаем решение
x=-arcsin(0.5[a+[$8730$](a²+8)])+2 pi n (n - целое)

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