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Ecuación x/4=x/9
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Ecuación x^3=4x^2+5x
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Ecuación x^2=8x
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Ecuación x^2=-x+6
- Expresar {x} en función de y en la ecuación:
- -8*x-6*y=-20
- -20*x+1*y=12
- -6*x-8*y=18
- -11*x-18*y=18
- Suma de la serie:
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0,01
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Expresiones idénticas
- tres ,1 dos x^ once -2,25x^ once - cero ,88x^ once = cero , uno
- 3,12x en el grado 11 menos 2,25x en el grado 11 menos 0,88x en el grado 11 es igual a 0,01
- tres ,1 dos x en el grado once menos 2,25x en el grado once menos cero ,88x en el grado once es igual a cero , uno
- 3,12x11-2,25x11-0,88x11=0,01
- 3,12x^11-2,25x^11-0,88x^11=O,01
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Expresiones semejantes
- 3,12x^11+2,25x^11-0,88x^11=0,01
- 3,12x^11-2,25x^11+0,88x^11=0,01
3,12x^11-2,25x^11-0,88x^11=0,01 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
11 11 11 78*x 9*x 22*x ------ - ----- - ------ = 1/100 25 4 25
$$- \frac{22 x^{11}}{25} + \left(- \frac{9 x^{11}}{4} + \frac{78 x^{11}}{25}\right) = \frac{1}{100}$$
Solución detallada
Tenemos la ecuación
$$- \frac{22 x^{11}}{25} + \left(- \frac{9 x^{11}}{4} + \frac{78 x^{11}}{25}\right) = \frac{1}{100}$$
Ya que la potencia en la ecuación es igual a = 11 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 11 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[11]{- \frac{1}{100}} \sqrt[11]{x^{11}} = \sqrt[11]{\frac{1}{100}}$$
o
$$\frac{\sqrt[11]{-1} \cdot 10^{\frac{9}{11}} x}{10} = \frac{10^{\frac{9}{11}}}{10}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
Abrimos los paréntesis en el miembro derecho de la ecuación
Dividamos ambos miembros de la ecuación en (-1)^(1/11)*10^(9/11)/10
Obtenemos la respuesta: x = -(-1)^(10/11)
Las demás 10 raíces son complejas.
hacemos el cambio:
$$z = x$$
entonces la ecuación será así:
$$z^{11} = -1$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{11} e^{11 i p} = -1$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{11 i p} = 1$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(11 p \right)} + \cos{\left(11 p \right)} = 1$$
es decir
$$\cos{\left(11 p \right)} = 1$$
y
$$\sin{\left(11 p \right)} = 0$$
entonces
$$p = \frac{2 \pi N}{11}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}$$
$$z_{2} = - \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}$$
$$z_{3} = - \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{4} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{5} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} - i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}$$
$$z_{6} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{7} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} - i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}$$
$$z_{8} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{9} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{10} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}$$
$$z_{11} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}$$
hacemos cambio inverso
$$z = x$$
$$x = z$$
Entonces la respuesta definitiva es:
$$x_{1} = \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}$$
$$x_{2} = - \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}$$
$$x_{3} = - \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{4} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{5} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} - i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}$$
$$x_{6} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{7} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} - i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}$$
$$x_{8} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{9} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{10} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}$$
$$x_{11} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}$$
$$- \frac{22 x^{11}}{25} + \left(- \frac{9 x^{11}}{4} + \frac{78 x^{11}}{25}\right) = \frac{1}{100}$$
Ya que la potencia en la ecuación es igual a = 11 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 11 de las dos partes de la ecuación:
Obtenemos:
$$\sqrt[11]{- \frac{1}{100}} \sqrt[11]{x^{11}} = \sqrt[11]{\frac{1}{100}}$$
o
$$\frac{\sqrt[11]{-1} \cdot 10^{\frac{9}{11}} x}{10} = \frac{10^{\frac{9}{11}}}{10}$$
Abrimos los paréntesis en el miembro izquierdo de la ecuación
x-1^1/11*10^9/11/10 = 10^(9/11)/10
Abrimos los paréntesis en el miembro derecho de la ecuación
x-1^1/11*10^9/11/10 = 10^9/11/10
Dividamos ambos miembros de la ecuación en (-1)^(1/11)*10^(9/11)/10
x = 10^(9/11)/10 / ((-1)^(1/11)*10^(9/11)/10)
Obtenemos la respuesta: x = -(-1)^(10/11)
Las demás 10 raíces son complejas.
hacemos el cambio:
$$z = x$$
entonces la ecuación será así:
$$z^{11} = -1$$
Cualquier número complejo se puede presentar que:
$$z = r e^{i p}$$
sustituimos en la ecuación
$$r^{11} e^{11 i p} = -1$$
donde
$$r = 1$$
- módulo del número complejo
Sustituyamos r:
$$e^{11 i p} = 1$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(11 p \right)} + \cos{\left(11 p \right)} = 1$$
es decir
$$\cos{\left(11 p \right)} = 1$$
y
$$\sin{\left(11 p \right)} = 0$$
entonces
$$p = \frac{2 \pi N}{11}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
$$z_{1} = \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}$$
$$z_{2} = - \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}$$
$$z_{3} = - \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{4} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{5} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} - i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}$$
$$z_{6} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{7} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} - i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}$$
$$z_{8} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{9} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$z_{10} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}$$
$$z_{11} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}$$
hacemos cambio inverso
$$z = x$$
$$x = z$$
Entonces la respuesta definitiva es:
$$x_{1} = \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}$$
$$x_{2} = - \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}$$
$$x_{3} = - \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{4} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{5} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} - i \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}$$
$$x_{6} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{7} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} - i \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}$$
$$x_{8} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{9} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
$$x_{10} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} - i \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}$$
$$x_{11} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - i \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}$$
Suma y producto de raíces
[src]
suma
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\ / /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\ / /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\ / /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\ / /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\ /pi\ /pi\ 2/pi\ 2/pi\ /pi\ /pi\
- 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*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----| + I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----| + I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----| + I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----| + I*sin|--| + cos|--| + sin |--| - cos |--| - 2*I*cos|--|*sin|--|
\11/ \11/ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 / \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 / \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 / \11/ \11/ \11/ \11/ \11/ \11/
$$\left(- \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right) + \left(\left(\left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}\right)\right) + \left(\left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}\right)\right) + \left(\left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right) + \left(\left(\left(\sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(- \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}\right)\right) + \left(\left(\left(\sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(- \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}\right)\right) + \left(\left(- \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}\right) + \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right)\right)\right) + \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right)\right)\right) + \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right)\right)\right)\right)\right) + \left(\cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}\right)\right)$$
=
2/pi\ / /pi\ /2*pi\ /2*pi\ /pi\\ / /pi\ /3*pi\ /3*pi\ /pi\\ / /pi\ /4*pi\ /4*pi\ /pi\\ / /pi\ /5*pi\ /5*pi\ /pi\\ / /2*pi\ /pi\ /pi\ /2*pi\\ / /4*pi\ /pi\ /pi\ /4*pi\\ / /pi\ /3*pi\ /3*pi\ /pi\\ / /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /pi\ /3*pi\ /pi\ /5*pi\ /pi\ /2*pi\ /pi\ /4*pi\ /pi\ /pi\ /pi\
- 2*cos |--| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + I*|cos|--|*sin|----| - cos|----|*sin|--|| + I*|cos|--|*sin|----| + cos|----|*sin|--|| + I*|cos|--|*sin|----| - cos|----|*sin|--|| + 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*cos|--|*cos|----| + 2*cos|--|*cos|----| - 2*I*cos|--|*sin|--| + cos|--|
\11/ \ \11/ \ 11 / \ 11 / \11// \ \11/ \ 11 / \ 11 / \11// \ \11/ \ 11 / \ 11 / \11// \ \11/ \ 11 / \ 11 / \11// \ \ 11 / \11/ \11/ \ 11 // \ \ 11 / \11/ \11/ \ 11 // \ \11/ \ 11 / \ 11 / \11// \ \11/ \ 11 / \ 11 / \11// \11/ \11/ \ 11 / \11/ \ 11 / \11/ \ 11 / \11/ \ 11 / \11/ \11/ \11/
$$- 2 \cos^{2}{\left(\frac{\pi}{11} \right)} - 2 \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - 2 \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + 2 \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} + 2 \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(- \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}\right) + i \left(- \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}\right) + i \left(- \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}\right) - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + i \left(- \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}\right) + i \sin{\left(\frac{\pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right) + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right) + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right) + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
producto
/ 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\\ / / /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\\ / / /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\\ / / /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\\ / / /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\\ / /pi\ /pi\\ / 2/pi\ 2/pi\ /pi\ /pi\\ |- 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*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----||*|I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----||*|I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----||*|I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----||*|I*sin|--| + cos|--||*|sin |--| - cos |--| - 2*I*cos|--|*sin|--|| \ \11/ \11// \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 // \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 // \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 // \ \11/ \11// \ \11/ \11/ \11/ \11//
$$\left(- \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}\right) \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right) \left(\sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(- \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}\right)\right) \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right) \left(\sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(- \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}\right)\right) \left(- \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}\right)\right) \left(- \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}\right)\right) \left(\cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}\right) \left(- \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
=
2/pi\ 6/pi\ 6/pi\ 4/pi\
3 - 12*sin |--| - 4*cos |--| - 4*sin |--| + 12*sin |--|
\11/ \11/ \11/ \11/
$$- 4 \cos^{6}{\left(\frac{\pi}{11} \right)} - 12 \sin^{2}{\left(\frac{\pi}{11} \right)} - 4 \sin^{6}{\left(\frac{\pi}{11} \right)} + 12 \sin^{4}{\left(\frac{\pi}{11} \right)} + 3$$
3 - 12*sin(pi/11)^2 - 4*cos(pi/11)^6 - 4*sin(pi/11)^6 + 12*sin(pi/11)^4
Respuesta rápida
[src]
2/pi\ 2/pi\
x1 = - cos |--| - sin |--|
\11/ \11/
$$x_{1} = - \cos^{2}{\left(\frac{\pi}{11} \right)} - \sin^{2}{\left(\frac{\pi}{11} \right)}$$
/ /pi\ /2*pi\ /2*pi\ /pi\\ /pi\ /2*pi\ /pi\ /2*pi\
x2 = I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{2} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
/ /2*pi\ /pi\ /pi\ /2*pi\\ /pi\ /2*pi\ /pi\ /2*pi\
x3 = I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----|
\ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 /
$$x_{3} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{2 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)} + i \left(- \sin{\left(\frac{2 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{2 \pi}{11} \right)}\right)$$
/ /pi\ /4*pi\ /4*pi\ /pi\\ /pi\ /4*pi\ /pi\ /4*pi\
x4 = I*|cos|--|*sin|----| + cos|----|*sin|--|| + cos|--|*cos|----| - sin|--|*sin|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{4} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(\sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
/ /4*pi\ /pi\ /pi\ /4*pi\\ /pi\ /4*pi\ /pi\ /4*pi\
x5 = I*|cos|----|*sin|--| - cos|--|*sin|----|| + cos|--|*cos|----| + sin|--|*sin|----|
\ \ 11 / \11/ \11/ \ 11 // \11/ \ 11 / \11/ \ 11 /
$$x_{5} = \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{4 \pi}{11} \right)} + \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} + i \left(- \sin{\left(\frac{4 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{4 \pi}{11} \right)}\right)$$
/ /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\
x6 = I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{6} = - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
/ /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\
x7 = I*|cos|--|*sin|----| - cos|----|*sin|--|| - cos|--|*cos|----| - sin|--|*sin|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{7} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}\right)$$
/ /pi\ /3*pi\ /3*pi\ /pi\\ /pi\ /3*pi\ /pi\ /3*pi\
x8 = I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{8} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{3 \pi}{11} \right)} + i \left(- \sin{\left(\frac{3 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{3 \pi}{11} \right)}\right)$$
/ /pi\ /5*pi\ /5*pi\ /pi\\ /pi\ /5*pi\ /pi\ /5*pi\
x9 = I*|- cos|--|*sin|----| - cos|----|*sin|--|| + sin|--|*sin|----| - cos|--|*cos|----|
\ \11/ \ 11 / \ 11 / \11// \11/ \ 11 / \11/ \ 11 /
$$x_{9} = - \cos{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)} + \sin{\left(\frac{\pi}{11} \right)} \sin{\left(\frac{5 \pi}{11} \right)} + i \left(- \sin{\left(\frac{5 \pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)} - \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{5 \pi}{11} \right)}\right)$$
/pi\ /pi\
x10 = I*sin|--| + cos|--|
\11/ \11/
$$x_{10} = \cos{\left(\frac{\pi}{11} \right)} + i \sin{\left(\frac{\pi}{11} \right)}$$
2/pi\ 2/pi\ /pi\ /pi\
x11 = sin |--| - cos |--| - 2*I*cos|--|*sin|--|
\11/ \11/ \11/ \11/
$$x_{11} = - \cos^{2}{\left(\frac{\pi}{11} \right)} + \sin^{2}{\left(\frac{\pi}{11} \right)} - 2 i \sin{\left(\frac{\pi}{11} \right)} \cos{\left(\frac{\pi}{11} \right)}$$
x11 = -cos(pi/11)^2 + sin(pi/11)^2 - 2*i*sin(pi/11)*cos(pi/11)
Respuesta numérica
[src]
x1 = 0.959492973614497 + 0.28173255684143*i
x2 = -0.415415013001886 - 0.909631995354518*i
x3 = 0.142314838273285 + 0.989821441880933*i
x4 = 0.654860733945285 - 0.755749574354258*i
x5 = 0.959492973614497 - 0.28173255684143*i
x6 = -1.0
x7 = -0.415415013001886 + 0.909631995354518*i
x8 = 0.142314838273285 - 0.989821441880933*i
x9 = 0.654860733945285 + 0.755749574354258*i
x10 = -0.841253532831181 + 0.540640817455598*i
x11 = -0.841253532831181 - 0.540640817455598*i
x11 = -0.841253532831181 - 0.540640817455598*i