x^9=-1 (уравнение)

Найду корень уравнения: x^9=-1

Вы ввели

 9     
x  = -1

$$x^{9} = -1$$

Подробное решение

Дано уравнение
$$x^{9} = -1$$
Т.к. степень в ур-нии равна = 9 - не содержит чётного числа в числителе, то
ур-ние будет иметь один действительный корень.
Извлечём корень 9-й степени из обеих частей ур-ния:
Получим:
$$\sqrt[9]{\left(1 x + 0\right)^{9}} = \sqrt[9]{-1}$$
или
$$x = \sqrt[9]{-1}$$
Раскрываем скобочки в правой части ур-ния

x = -1^1/9

Получим ответ: x = (-1)^(1/9)

Остальные 8 корня(ей) являются комплексными.
сделаем замену:
$$z = x$$
тогда ур-ние будет таким:
$$z^{9} = -1$$
Любое комплексное число можно представить так:
$$z = r e^{i p}$$
подставляем в уравнение
$$r^{9} e^{9 i p} = -1$$
где
$$r = 1$$
- модуль комплексного числа
Подставляем r:
$$e^{9 i p} = -1$$
Используя формулу Эйлера, найдём корни для p
$$i \sin{\left(9 p \right)} + \cos{\left(9 p \right)} = -1$$
значит
$$\cos{\left(9 p \right)} = -1$$
и
$$\sin{\left(9 p \right)} = 0$$
тогда
$$p = \frac{2 \pi N}{9} + \frac{\pi}{9}$$
где N=0,1,2,3,...
Перебирая значения N и подставив p в формулу для z
Значит, решением будет для z:
$$z_{1} = \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)}$$
$$z_{2} = - \cos^{2}{\left(\frac{\pi}{9} \right)} - \sin^{2}{\left(\frac{\pi}{9} \right)}$$
$$z_{3} = - \cos^{2}{\left(\frac{\pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{4} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{5} = \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}$$
$$z_{6} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$z_{7} = \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} - i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}$$
$$z_{8} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}}{2}$$
$$z_{9} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{9} \right)}}{2}$$
делаем обратную замену
$$z = x$$
$$x = z$$

Тогда, окончательный ответ:
$$x_{1} = \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)}$$
$$x_{2} = - \cos^{2}{\left(\frac{\pi}{9} \right)} - \sin^{2}{\left(\frac{\pi}{9} \right)}$$
$$x_{3} = - \cos^{2}{\left(\frac{\pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{4} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{5} = \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - i \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}$$
$$x_{6} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$
$$x_{7} = \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} - i \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}$$
$$x_{8} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}}{2}$$
$$x_{9} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} i \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{i \sin{\left(\frac{\pi}{9} \right)}}{2}$$

Быстрый ответ

          2/pi\      2/pi\
x1 = - cos |--| - sin |--|
           \9 /       \9 /

$$x_{1} = - \cos^{2}{\left(\frac{\pi}{9} \right)} - \sin^{2}{\left(\frac{\pi}{9} \right)}$$

       /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\
x2 = I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----|
       \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  /

$$x_{2} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

       /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\
x3 = I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----|
       \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  /

$$x_{3} = \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right)$$

       /   /pi\    /4*pi\      /4*pi\    /pi\\      /pi\    /4*pi\      /pi\    /4*pi\
x4 = I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----|
       \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  /

$$x_{4} = - \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

       /   /4*pi\    /pi\      /pi\    /4*pi\\      /pi\    /4*pi\      /pi\    /4*pi\
x5 = I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----|
       \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  /

$$x_{5} = \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + i \left(- \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right)$$

          /pi\     /     /pi\     ___    /pi\\     ___    /pi\
       cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--|
          \9 /     |     \9 /            \9 /|            \9 /
x6 = - ------- + I*|- ------- + -------------| - -------------
          2        \     2            2      /         2      

$$x_{6} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$

          /pi\     /     /pi\     ___    /pi\\     ___    /pi\
       cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--|
          \9 /     |     \9 /            \9 /|            \9 /
x7 = - ------- + I*|- ------- - -------------| + -------------
          2        \     2            2      /         2      

$$x_{7} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2}\right)$$

          /pi\      /pi\
x8 = I*sin|--| + cos|--|
          \9 /      \9 /

$$x_{8} = \cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)}$$

        2/pi\      2/pi\          /pi\    /pi\
x9 = sin |--| - cos |--| - 2*I*cos|--|*sin|--|
         \9 /       \9 /          \9 /    \9 /

$$x_{9} = - \cos^{2}{\left(\frac{\pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}$$

Сумма и произведение корней

сумма

                                                                                                                                                                                                                                                                                                                                                                                 /pi\     /     /pi\     ___    /pi\\     ___    /pi\        /pi\     /     /pi\     ___    /pi\\     ___    /pi\                                                                  
                                                                                                                                                                                                                                                                                                                                                                              cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--|     cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--|                                                                  
         2/pi\      2/pi\     /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\     /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\     /   /pi\    /4*pi\      /4*pi\    /pi\\      /pi\    /4*pi\      /pi\    /4*pi\     /   /4*pi\    /pi\      /pi\    /4*pi\\      /pi\    /4*pi\      /pi\    /4*pi\        \9 /     |     \9 /            \9 /|            \9 /        \9 /     |     \9 /            \9 /|            \9 /        /pi\      /pi\      2/pi\      2/pi\          /pi\    /pi\
0 + - cos |--| - sin |--| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----| + - ------- + I*|- ------- + -------------| - ------------- + - ------- + I*|- ------- - -------------| + ------------- + I*sin|--| + cos|--| + sin |--| - cos |--| - 2*I*cos|--|*sin|--|
          \9 /       \9 /     \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  /     \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  /     \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  /     \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  /        2        \     2            2      /         2              2        \     2            2      /         2              \9 /      \9 /       \9 /       \9 /          \9 /    \9 /

$$\left(- \cos^{2}{\left(\frac{\pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - \left(- 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} - 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} + \cos^{2}{\left(\frac{\pi}{9} \right)} - i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) - i \left(- \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) - i \sin{\left(\frac{\pi}{9} \right)} - i \left(- \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right) - i \left(- \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right) - i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2}\right)\right)$$

=

                 /     /pi\     ___    /pi\\     /     /pi\     ___    /pi\\                                                                                                                                                                                                                                                              
                 |  sin|--|   \/ 3 *cos|--||     |  sin|--|   \/ 3 *cos|--||                                                                                                                                                                                                                                                              
       2/pi\     |     \9 /            \9 /|     |     \9 /            \9 /|     /   /pi\    /2*pi\      /2*pi\    /pi\\     /   /pi\    /4*pi\      /4*pi\    /pi\\     /   /2*pi\    /pi\      /pi\    /2*pi\\     /   /4*pi\    /pi\      /pi\    /4*pi\\        /pi\        /pi\    /2*pi\        /pi\    /4*pi\          /pi\    /pi\
- 2*cos |--| + I*|- ------- + -------------| + I*|- ------- - -------------| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + I*|cos|----|*sin|--| - cos|--|*sin|----|| + I*sin|--| + 2*cos|--|*cos|----| + 2*cos|--|*cos|----| - 2*I*cos|--|*sin|--|
        \9 /     \     2            2      /     \     2            2      /     \   \9 /    \ 9  /      \ 9  /    \9 //     \   \9 /    \ 9  /      \ 9  /    \9 //     \   \ 9  /    \9 /      \9 /    \ 9  //     \   \ 9  /    \9 /      \9 /    \ 9  //        \9 /        \9 /    \ 9  /        \9 /    \ 9  /          \9 /    \9 /

$$- 2 \cos^{2}{\left(\frac{\pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + 2 \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(- \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right) - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + i \left(- \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right) + i \sin{\left(\frac{\pi}{9} \right)} + i \left(- \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right) + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

произведение

                                                                                                                                                                                                                                                                                                                                                                          /     /pi\     /     /pi\     ___    /pi\\     ___    /pi\\ /     /pi\     /     /pi\     ___    /pi\\     ___    /pi\\                                                                  
                                                                                                                                                                                                                                                                                                                                                                          |  cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--|| |  cos|--|     |  sin|--|   \/ 3 *cos|--||   \/ 3 *sin|--||                                                                  
  /     2/pi\      2/pi\\ /  /   /pi\    /2*pi\      /2*pi\    /pi\\      /pi\    /2*pi\      /pi\    /2*pi\\ /  /   /2*pi\    /pi\      /pi\    /2*pi\\      /pi\    /2*pi\      /pi\    /2*pi\\ /  /   /pi\    /4*pi\      /4*pi\    /pi\\      /pi\    /4*pi\      /pi\    /4*pi\\ /  /   /4*pi\    /pi\      /pi\    /4*pi\\      /pi\    /4*pi\      /pi\    /4*pi\\ |     \9 /     |     \9 /            \9 /|            \9 /| |     \9 /     |     \9 /            \9 /|            \9 /| /     /pi\      /pi\\ /   2/pi\      2/pi\          /pi\    /pi\\
1*|- cos |--| - sin |--||*|I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----||*|I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----||*|I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----||*|I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----||*|- ------- + I*|- ------- + -------------| - -------------|*|- ------- + I*|- ------- - -------------| + -------------|*|I*sin|--| + cos|--||*|sin |--| - cos |--| - 2*I*cos|--|*sin|--||
  \      \9 /       \9 // \  \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  // \  \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  // \  \   \9 /    \ 9  /      \ 9  /    \9 //      \9 /    \ 9  /      \9 /    \ 9  // \  \   \ 9  /    \9 /      \9 /    \ 9  //      \9 /    \ 9  /      \9 /    \ 9  // \     2        \     2            2      /         2      / \     2        \     2            2      /         2      / \     \9 /      \9 // \    \9 /       \9 /          \9 /    \9 //

$$1 \left(- \cos^{2}{\left(\frac{\pi}{9} \right)} - \sin^{2}{\left(\frac{\pi}{9} \right)}\right) \left(- \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(\sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{2 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)} + i \left(- \sin{\left(\frac{2 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{2 \pi}{9} \right)}\right)\right) \left(- \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + i \left(\sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)\right) \left(\cos{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \sin{\left(\frac{4 \pi}{9} \right)} + i \left(- \sin{\left(\frac{4 \pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)} + \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{4 \pi}{9} \right)}\right)\right) \left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(\cos{\left(\frac{\pi}{9} \right)} + i \sin{\left(\frac{\pi}{9} \right)}\right) \left(- \cos^{2}{\left(\frac{\pi}{9} \right)} + \sin^{2}{\left(\frac{\pi}{9} \right)} - 2 i \sin{\left(\frac{\pi}{9} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)$$

=

                                                                /              /    ___        /pi\        /pi\    /pi\\\                                                    2*pi*I
                                                                |            I*|- \/ 3  + 2*cos|--| + 4*cos|--|*cos|--|||                                                    ------
/        ___\ /      /    ___        /pi\        /pi\    /pi\\\ |     /pi\     \               \18/        \9 /    \18//| /     /pi\      /pi\\ /       /2*pi\      /5*pi\\    9   
\1 + I*\/ 3 /*|2 + I*|- \/ 3  + 2*cos|--| - 4*cos|--|*cos|--|||*|- sin|--| + -------------------------------------------|*|I*cos|--| + sin|--||*|- I*sin|----| + sin|----||*e      
              \      \               \18/        \9 /    \18/// \     \18/                        4                     / \     \18/      \18// \       \ 9  /      \ 18 //        
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$$\frac{\left(1 + \sqrt{3} i\right) \left(2 + i \left(- 4 \cos{\left(\frac{\pi}{18} \right)} \cos{\left(\frac{\pi}{9} \right)} - \sqrt{3} + 2 \cos{\left(\frac{\pi}{18} \right)}\right)\right) \left(- \sin{\left(\frac{\pi}{18} \right)} + \frac{i \left(- \sqrt{3} + 2 \cos{\left(\frac{\pi}{18} \right)} + 4 \cos{\left(\frac{\pi}{18} \right)} \cos{\left(\frac{\pi}{9} \right)}\right)}{4}\right) \left(\sin{\left(\frac{\pi}{18} \right)} + i \cos{\left(\frac{\pi}{18} \right)}\right) \left(\sin{\left(\frac{5 \pi}{18} \right)} - i \sin{\left(\frac{2 \pi}{9} \right)}\right) e^{\frac{2 i \pi}{9}}}{8}$$

Численный ответ

x1 = -0.17364817766693 - 0.984807753012208*i

x2 = 0.939692620785908 - 0.342020143325669*i

x3 = 0.5 - 0.866025403784439*i

x4 = -0.766044443118978 - 0.642787609686539*i

x5 = -1.0

x6 = 0.5 + 0.866025403784439*i

x7 = -0.766044443118978 + 0.642787609686539*i

x8 = -0.17364817766693 + 0.984807753012208*i

x9 = 0.939692620785908 + 0.342020143325669*i

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