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Ecuación z^2-6*z+10=0
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Expresiones idénticas
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- z en el grado 12 es igual a raíz cuadrada de (3) menos i
- z en el grado doce es igual a raíz cuadrada de (tres) menos i
- z^12=√(3)-i
- z12=sqrt(3)-i
- z12=sqrt3-i
- z^12=sqrt3-i
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Expresiones semejantes
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- z^5-sqrt3-i=0
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x)=1
- sqrt(x-2)+sqrt(y-2)=2
- sqrt(x^2+9)=2*x-3
- sqrt(x)=-2
- sqrt(2-x)+sqrt(-x-1)=sqrt(-5*x-7)
z^12=sqrt(3)-i la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$z^{12} = \sqrt{3} - i$$
Ya que la potencia en la ecuación es igual a = 12 y miembro libre = sqrt(3) - i complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 12 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{12} = \sqrt{3} - i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{12} e^{12 i p} = \sqrt{3} - i$$
donde
$$r = \sqrt[12]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{12 i p} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(12 p \right)} + \cos{\left(12 p \right)} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
es decir
$$\cos{\left(12 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(12 p \right)} = - \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{6} - \frac{\pi}{72}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:
$$w_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$w_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$w_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$w_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$w_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$z_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$z_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$z_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$z_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z^{12} = \sqrt{3} - i$$
Ya que la potencia en la ecuación es igual a = 12 y miembro libre = sqrt(3) - i complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 12 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{12} = \sqrt{3} - i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{12} e^{12 i p} = \sqrt{3} - i$$
donde
$$r = \sqrt[12]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{12 i p} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(12 p \right)} + \cos{\left(12 p \right)} = \frac{\sqrt{3}}{2} - \frac{i}{2}$$
es decir
$$\cos{\left(12 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(12 p \right)} = - \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{6} - \frac{\pi}{72}$$
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:
$$w_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$w_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$w_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$w_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$w_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$w_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$z_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
$$z_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$z_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
$$z_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2}$$
$$z_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} i \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}}{2}$$
Gráfica
Respuesta rápida
[src]
12___ /pi\ 12___ /pi\
z1 = - \/ 2 *sin|--| - I*\/ 2 *cos|--|
\72/ \72/
$$z_{1} = - \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
12___ /pi\ 12___ /pi\
z2 = \/ 2 *sin|--| + I*\/ 2 *cos|--|
\72/ \72/
$$z_{2} = \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}$$
12___ /pi\ 12___ /pi\
z3 = - \/ 2 *cos|--| + I*\/ 2 *sin|--|
\72/ \72/
$$z_{3} = - \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
12___ /pi\ 12___ /pi\
z4 = \/ 2 *cos|--| - I*\/ 2 *sin|--|
\72/ \72/
$$z_{4} = \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}$$
/ 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|
| \72/ \72/| \72/ \72/
z5 = I*|- ------------- - -------------------| - ------------- + -------------------
\ 2 2 / 2 2
$$z_{5} = - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
|\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|
| \72/ \72/| \72/ \72/
z6 = I*|------------- - -------------------| + ------------- + -------------------
\ 2 2 / 2 2
$$z_{6} = \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
|\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \72/ \72/| \72/ \72/
z7 = I*|------------- - -------------------| - ------------- - -------------------
\ 2 2 / 2 2
$$z_{7} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
|\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \72/ \72/| \72/ \72/
z8 = I*|------------- + -------------------| - ------------- + -------------------
\ 2 2 / 2 2
$$z_{8} = - \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/ 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \72/ \72/| \72/ \72/
z9 = I*|- ------------- - -------------------| + ------------- - -------------------
\ 2 2 / 2 2
$$z_{9} = - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/ 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \72/ \72/| \72/ \72/
z10 = I*|- ------------- + -------------------| + ------------- + -------------------
\ 2 2 / 2 2
$$z_{10} = \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/ 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|
| \72/ \72/| \72/ \72/
z11 = I*|- ------------- + -------------------| - ------------- - -------------------
\ 2 2 / 2 2
$$z_{11} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)$$
/12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
|\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|
| \72/ \72/| \72/ \72/
z12 = I*|------------- + -------------------| + ------------- - -------------------
\ 2 2 / 2 2
$$z_{12} = - \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)$$
z12 = -2^(1/12)*sqrt(3)*cos(pi/72)/2 + 2^(1/12)*sin(pi/72)/2 + i*(2^(1/12)*sqrt(3)*sin(pi/72)/2 + 2^(1/12)*cos(pi/72)/2)
Suma y producto de raíces
[src]
suma
/ 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\ /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\
| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--| |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--| |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--| |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--| | \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--| |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|
12___ /pi\ 12___ /pi\ 12___ /pi\ 12___ /pi\ 12___ /pi\ 12___ /pi\ 12___ /pi\ 12___ /pi\ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/ | \72/ \72/| \72/ \72/
- \/ 2 *sin|--| - I*\/ 2 *cos|--| + \/ 2 *sin|--| + I*\/ 2 *cos|--| + - \/ 2 *cos|--| + I*\/ 2 *sin|--| + \/ 2 *cos|--| - I*\/ 2 *sin|--| + I*|- ------------- - -------------------| - ------------- + ------------------- + I*|------------- - -------------------| + ------------- + ------------------- + I*|------------- - -------------------| - ------------- - ------------------- + I*|------------- + -------------------| - ------------- + ------------------- + I*|- ------------- - -------------------| + ------------- - ------------------- + I*|- ------------- + -------------------| + ------------- + ------------------- + I*|- ------------- + -------------------| - ------------- - ------------------- + I*|------------- + -------------------| + ------------- - -------------------
\72/ \72/ \72/ \72/ \72/ \72/ \72/ \72/ \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(\left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) + \left(\left(\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) + \left(\left(\left(- \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) + \left(\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right)\right) + \left(- \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right)\right)\right)\right) + \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)$$
=
/12___ /pi\ 12___ ___ /pi\\ /12___ /pi\ 12___ ___ /pi\\ /12___ /pi\ 12___ ___ /pi\\ /12___ /pi\ 12___ ___ /pi\\ / 12___ /pi\ 12___ ___ /pi\\ / 12___ /pi\ 12___ ___ /pi\\ / 12___ /pi\ 12___ ___ /pi\\ / 12___ /pi\ 12___ ___ /pi\\ |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \72/ \72/| | \72/ \72/| | \72/ \72/| | \72/ \72/| | \72/ \72/| | \72/ \72/| | \72/ \72/| | \72/ \72/| I*|------------- + -------------------| + I*|------------- - -------------------| + I*|------------- + -------------------| + I*|------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right) + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)$$
producto
/ / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / / 12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\ / /12___ /pi\ 12___ ___ /pi\\ 12___ /pi\ 12___ ___ /pi\\
| | \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| | | \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | |\/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--||
/ 12___ /pi\ 12___ /pi\\ /12___ /pi\ 12___ /pi\\ / 12___ /pi\ 12___ /pi\\ /12___ /pi\ 12___ /pi\\ | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/| | | \72/ \72/| \72/ \72/|
|- \/ 2 *sin|--| - I*\/ 2 *cos|--||*|\/ 2 *sin|--| + I*\/ 2 *cos|--||*|- \/ 2 *cos|--| + I*\/ 2 *sin|--||*|\/ 2 *cos|--| - I*\/ 2 *sin|--||*|I*|- ------------- - -------------------| - ------------- + -------------------|*|I*|------------- - -------------------| + ------------- + -------------------|*|I*|------------- - -------------------| - ------------- - -------------------|*|I*|------------- + -------------------| - ------------- + -------------------|*|I*|- ------------- - -------------------| + ------------- - -------------------|*|I*|- ------------- + -------------------| + ------------- + -------------------|*|I*|- ------------- + -------------------| - ------------- - -------------------|*|I*|------------- + -------------------| + ------------- - -------------------|
\ \72/ \72// \ \72/ \72// \ \72/ \72// \ \72/ \72// \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(- \sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) \left(\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \cos{\left(\frac{\pi}{72} \right)}\right) \left(- \sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} + \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) \left(\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)} - \sqrt[12]{2} i \sin{\left(\frac{\pi}{72} \right)}\right) \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} - \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(- \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2}\right)\right) \left(- \frac{\sqrt[12]{2} \sqrt{3} \cos{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \sin{\left(\frac{\pi}{72} \right)}}{2} + i \left(\frac{\sqrt[12]{2} \sqrt{3} \sin{\left(\frac{\pi}{72} \right)}}{2} + \frac{\sqrt[12]{2} \cos{\left(\frac{\pi}{72} \right)}}{2}\right)\right)$$
=
-pi*I
2 2 2 2 2 ------
/ /11*pi\ /11*pi\\ / /23*pi\ /13*pi\\ / /pi\ /pi\\ / /13*pi\ /13*pi\\ / /11*pi\ /25*pi\\ 36
2*|- sin|-----| + I*cos|-----|| *|- sin|-----| + I*sin|-----|| *|I*cos|--| + sin|--|| *|I*cos|-----| + sin|-----|| *|I*sin|-----| + sin|-----|| *e
\ \ 72 / \ 72 // \ \ 72 / \ 72 // \ \72/ \72// \ \ 72 / \ 72 // \ \ 72 / \ 72 //
$$2 \left(\sin{\left(\frac{\pi}{72} \right)} + i \cos{\left(\frac{\pi}{72} \right)}\right)^{2} \left(- \sin{\left(\frac{11 \pi}{72} \right)} + i \cos{\left(\frac{11 \pi}{72} \right)}\right)^{2} \left(\sin{\left(\frac{13 \pi}{72} \right)} + i \cos{\left(\frac{13 \pi}{72} \right)}\right)^{2} \left(- \sin{\left(\frac{23 \pi}{72} \right)} + i \sin{\left(\frac{13 \pi}{72} \right)}\right)^{2} \left(\sin{\left(\frac{25 \pi}{72} \right)} + i \sin{\left(\frac{11 \pi}{72} \right)}\right)^{2} e^{- \frac{i \pi}{36}}$$
2*(-sin(11*pi/72) + i*cos(11*pi/72))^2*(-sin(23*pi/72) + i*sin(13*pi/72))^2*(i*cos(pi/72) + sin(pi/72))^2*(i*cos(13*pi/72) + sin(13*pi/72))^2*(i*sin(11*pi/72) + sin(25*pi/72))^2*exp(-pi*i/36)
Respuesta numérica
[src]
z1 = 0.0462131311121356 + 1.05845472025127*i
z2 = -0.489205614594103 + 0.939755242049215*i
z3 = -0.569249105657163 - 0.89354211093708*i
z4 = -0.939755242049215 - 0.489205614594103*i
z5 = 0.569249105657163 + 0.89354211093708*i
z6 = -0.89354211093708 + 0.569249105657163*i
z7 = -0.0462131311121356 - 1.05845472025127*i
z8 = -1.05845472025127 + 0.0462131311121356*i
z9 = 0.939755242049215 + 0.489205614594103*i
z10 = 0.89354211093708 - 0.569249105657163*i
z11 = 0.489205614594103 - 0.939755242049215*i
z12 = 1.05845472025127 - 0.0462131311121356*i
z12 = 1.05845472025127 - 0.0462131311121356*i