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Консультация № 184382
05.11.2011, 10:16
51.74 руб.
06.11.2011, 07:25
0
2
2
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Дано комплексное число z. Требуется:
1) записать его в алгебраической и тригонометрической формах;
2) найти все корни уравнения w3 + z = 0.
z = 3/(1 - i[$8730$]3)
Дано комплексное число z. Требуется:
1) записать его в алгебраической и тригонометрической формах;
2) найти все корни уравнения w3 + z = 0.
z = 3/(1 - i[$8730$]3)
Обсуждение
Неизвестный
05.11.2011, 14:20
общий
это ответ
Здравствуйте, Aleksandrkib!
1) Для того чтобы записать его в алгебраической форме домножим дробь на сопряженное 1+i*sqrt(3) и выделить реальную и мнимую части, получается
z=(3+3*sqrt(3)*i)/4=3/4+3*sqrt(3)*i/4
далее, для нахождения тригонометрической формы, находим модуль числа z
abs(z)=sqrt(3/4^2+(3*sqrt(3)/4)^2)=6/4
получается 6/4*(1/2+i*sqrt(3)/2)
то есть угол fi=pi/3
отсюда тригонометрическая форма числа z:
z=6/4*(cos(pi/3)+i*sin(pi/3))
2). w^3=-z
W^3=-3/4-i*3*sqrt(3)
число -z=6/4*(cos(-2*pi/3)+i*sin(-2*pi/3))
по формуле для нахождения корня из комплексного числа
получаем корни
w1=(6/4)^1/3*(cos(-2*pi/9)+i*sin(-2*pi/9))
w2=(6/4)^1/3*(cos(-4*pi/9)+i*sin(-4*pi/9))
w3=(6/4)^1/3*(cos(-10*pi/9)+i*sin(-10*pi/9))
1) Для того чтобы записать его в алгебраической форме домножим дробь на сопряженное 1+i*sqrt(3) и выделить реальную и мнимую части, получается
z=(3+3*sqrt(3)*i)/4=3/4+3*sqrt(3)*i/4
далее, для нахождения тригонометрической формы, находим модуль числа z
abs(z)=sqrt(3/4^2+(3*sqrt(3)/4)^2)=6/4
получается 6/4*(1/2+i*sqrt(3)/2)
то есть угол fi=pi/3
отсюда тригонометрическая форма числа z:
z=6/4*(cos(pi/3)+i*sin(pi/3))
2). w^3=-z
W^3=-3/4-i*3*sqrt(3)
число -z=6/4*(cos(-2*pi/3)+i*sin(-2*pi/3))
по формуле для нахождения корня из комплексного числа
получаем корни
w1=(6/4)^1/3*(cos(-2*pi/9)+i*sin(-2*pi/9))
w2=(6/4)^1/3*(cos(-4*pi/9)+i*sin(-4*pi/9))
w3=(6/4)^1/3*(cos(-10*pi/9)+i*sin(-10*pi/9))
Форма ответа
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