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- La ecuación:
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Ecuación x^2+5x=36
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Ecuación 9-7(x+3)=5-6x
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Ecuación 4x-5=3+2(x+1)
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Ecuación 6/(x-2)+5/x=3
- Expresar {x} en función de y en la ecuación:
- -14*x-12*y=-13
- -20*x+14*y=16
- -7*x+15*y=-11
- -14*x+5*y=3
- Gráfico de la función y =:
-
z^8
-
Expresiones idénticas
- z^ ocho = tres + tres *i
- z en el grado 8 es igual a 3 más 3 multiplicar por i
- z en el grado ocho es igual a tres más tres multiplicar por i
- z8=3+3*i
- z⁸=3+3*i
- z^8=3+3i
- z8=3+3i
-
Expresiones semejantes
- z^2-(3+3i)z+5i=0
- x^3+(3i*x^2)-(3*(1-2i)x)+10-5i=0
- z^8=3-3*i
- x=1/2*Ln(((-2*sqrt(3)+3i)*i+7)/(-(-2*sqrt(3)+3i)*i+7))
z^8=3+3*i la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$z^{8} = 3 + 3 i$$
Ya que la potencia en la ecuación es igual a = 8 y miembro libre = 3 + 3*i complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 8 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{8} = 3 + 3 i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{8} e^{8 i p} = 3 + 3 i$$
donde
$$r = \sqrt[16]{2} \sqrt[8]{3}$$
- módulo del número complejo
Sustituyamos r:
$$e^{8 i p} = \frac{\sqrt{2} \left(3 + 3 i\right)}{6}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = \frac{\sqrt{2} \left(3 + 3 i\right)}{6}$$
es decir
$$\cos{\left(8 p \right)} = \frac{\sqrt{2}}{2}$$
y
$$\sin{\left(8 p \right)} = \frac{\sqrt{2}}{2}$$
entonces
$$p = \frac{\pi N}{4} + \frac{\pi}{32}$$
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:
$$w_{1} = - \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{2} = \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{3} = - \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{4} = \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{5} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{6} = \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{7} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{8} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{2} = \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{3} = - \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{4} = \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{5} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{6} = \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{7} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{8} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$z^{8} = 3 + 3 i$$
Ya que la potencia en la ecuación es igual a = 8 y miembro libre = 3 + 3*i complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 8 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{8} = 3 + 3 i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{8} e^{8 i p} = 3 + 3 i$$
donde
$$r = \sqrt[16]{2} \sqrt[8]{3}$$
- módulo del número complejo
Sustituyamos r:
$$e^{8 i p} = \frac{\sqrt{2} \left(3 + 3 i\right)}{6}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = \frac{\sqrt{2} \left(3 + 3 i\right)}{6}$$
es decir
$$\cos{\left(8 p \right)} = \frac{\sqrt{2}}{2}$$
y
$$\sin{\left(8 p \right)} = \frac{\sqrt{2}}{2}$$
entonces
$$p = \frac{\pi N}{4} + \frac{\pi}{32}$$
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:
$$w_{1} = - \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{2} = \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$w_{3} = - \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{4} = \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$w_{5} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{6} = \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{7} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$w_{8} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{2} = \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
$$z_{3} = - \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{4} = \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
$$z_{5} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{6} = \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{7} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2}$$
$$z_{8} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}}{2}$$
Gráfica
Respuesta rápida
[src]
16___ 8 ___ /pi\ 16___ 8 ___ /pi\
z1 = - \/ 2 *\/ 3 *sin|--| + I*\/ 2 *\/ 3 *cos|--|
\32/ \32/
$$z_{1} = - \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
16___ 8 ___ /pi\ 16___ 8 ___ /pi\
z2 = \/ 2 *\/ 3 *sin|--| - I*\/ 2 *\/ 3 *cos|--|
\32/ \32/
$$z_{2} = \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}$$
16___ 8 ___ /pi\ 16___ 8 ___ /pi\
z3 = - \/ 2 *\/ 3 *cos|--| - I*\/ 2 *\/ 3 *sin|--|
\32/ \32/
$$z_{3} = - \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
16___ 8 ___ /pi\ 16___ 8 ___ /pi\
z4 = \/ 2 *\/ 3 *cos|--| + I*\/ 2 *\/ 3 *sin|--|
\32/ \32/
$$z_{4} = \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}$$
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\
|2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|
| \32/ \32/| \32/ \32/
z5 = I*|------------------- + -------------------| + ------------------- - -------------------
\ 2 2 / 2 2
$$z_{5} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\
|2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|
| \32/ \32/| \32/ \32/
z6 = I*|------------------- - -------------------| + ------------------- + -------------------
\ 2 2 / 2 2
$$z_{6} = \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\
|2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|
| \32/ \32/| \32/ \32/
z7 = I*|------------------- - -------------------| - ------------------- - -------------------
\ 2 2 / 2 2
$$z_{7} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\
| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|
| \32/ \32/| \32/ \32/
z8 = I*|- ------------------- - -------------------| + ------------------- - -------------------
\ 2 2 / 2 2
$$z_{8} = - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)$$
z8 = -2^(9/16)*3^(1/8)*cos(pi/32)/2 + 2^(9/16)*3^(1/8)*sin(pi/32)/2 + i*(-2^(9/16)*3^(1/8)*cos(pi/32)/2 - 2^(9/16)*3^(1/8)*sin(pi/32)/2)
Suma y producto de raíces
[src]
suma
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\
|2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--| |2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--| |2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--| | 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|
16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\ | \32/ \32/| \32/ \32/ | \32/ \32/| \32/ \32/ | \32/ \32/| \32/ \32/ | \32/ \32/| \32/ \32/
- \/ 2 *\/ 3 *sin|--| + I*\/ 2 *\/ 3 *cos|--| + \/ 2 *\/ 3 *sin|--| - I*\/ 2 *\/ 3 *cos|--| + - \/ 2 *\/ 3 *cos|--| - I*\/ 2 *\/ 3 *sin|--| + \/ 2 *\/ 3 *cos|--| + I*\/ 2 *\/ 3 *sin|--| + I*|------------------- + -------------------| + ------------------- - ------------------- + I*|------------------- - -------------------| + ------------------- + ------------------- + I*|------------------- - -------------------| - ------------------- - ------------------- + I*|- ------------------- - -------------------| + ------------------- - -------------------
\32/ \32/ \32/ \32/ \32/ \32/ \32/ \32/ \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right) + \left(\left(\left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}\right) + \left(\left(\sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}\right) + \left(- \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}\right)\right)\right) + \left(\sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}\right)\right) + \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right)\right)$$
=
/ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ |2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| |2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| |2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|| | 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| | \32/ \32/| | \32/ \32/| | \32/ \32/| | \32/ \32/| I*|------------------- + -------------------| + I*|------------------- - -------------------| + I*|------------------- - -------------------| + I*|- ------------------- - -------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right) + i \left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)$$
producto
/ / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ / / 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\ 9/16 8 ___ /pi\ 9/16 8 ___ /pi\\
| |2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| | |2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| | |2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| | | 2 *\/ 3 *cos|--| 2 *\/ 3 *sin|--|| 2 *\/ 3 *sin|--| 2 *\/ 3 *cos|--||
/ 16___ 8 ___ /pi\ 16___ 8 ___ /pi\\ /16___ 8 ___ /pi\ 16___ 8 ___ /pi\\ / 16___ 8 ___ /pi\ 16___ 8 ___ /pi\\ /16___ 8 ___ /pi\ 16___ 8 ___ /pi\\ | | \32/ \32/| \32/ \32/| | | \32/ \32/| \32/ \32/| | | \32/ \32/| \32/ \32/| | | \32/ \32/| \32/ \32/|
|- \/ 2 *\/ 3 *sin|--| + I*\/ 2 *\/ 3 *cos|--||*|\/ 2 *\/ 3 *sin|--| - I*\/ 2 *\/ 3 *cos|--||*|- \/ 2 *\/ 3 *cos|--| - I*\/ 2 *\/ 3 *sin|--||*|\/ 2 *\/ 3 *cos|--| + I*\/ 2 *\/ 3 *sin|--||*|I*|------------------- + -------------------| + ------------------- - -------------------|*|I*|------------------- - -------------------| + ------------------- + -------------------|*|I*|------------------- - -------------------| - ------------------- - -------------------|*|I*|- ------------------- - -------------------| + ------------------- - -------------------|
\ \32/ \32// \ \32/ \32// \ \32/ \32// \ \32/ \32// \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(- \sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}\right) \left(\sqrt[16]{2} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \cos{\left(\frac{\pi}{32} \right)}\right) \left(- \sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} - \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}\right) \left(\sqrt[16]{2} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)} + \sqrt[16]{2} \sqrt[8]{3} i \sin{\left(\frac{\pi}{32} \right)}\right) \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(\frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2}\right)\right) \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} + \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2} + i \left(- \frac{2^{\frac{9}{16}} \sqrt[8]{3} \cos{\left(\frac{\pi}{32} \right)}}{2} - \frac{2^{\frac{9}{16}} \sqrt[8]{3} \sin{\left(\frac{\pi}{32} \right)}}{2}\right)\right)$$
=
pi*I
2 2 2 ----
___ / /7*pi\ /7*pi\\ / /pi\ /pi\\ / /7*pi\ /9*pi\\ 16
3*\/ 2 *|I*cos|----| + sin|----|| *|- I*cos|--| + sin|--|| *|- I*sin|----| + sin|----|| *e
\ \ 32 / \ 32 // \ \32/ \32// \ \ 32 / \ 32 //
$$3 \sqrt{2} \left(\sin{\left(\frac{\pi}{32} \right)} - i \cos{\left(\frac{\pi}{32} \right)}\right)^{2} \left(\sin{\left(\frac{7 \pi}{32} \right)} + i \cos{\left(\frac{7 \pi}{32} \right)}\right)^{2} \left(\sin{\left(\frac{9 \pi}{32} \right)} - i \sin{\left(\frac{7 \pi}{32} \right)}\right)^{2} e^{\frac{i \pi}{16}}$$
3*sqrt(2)*(i*cos(7*pi/32) + sin(7*pi/32))^2*(-i*cos(pi/32) + sin(pi/32))^2*(-i*sin(7*pi/32) + sin(9*pi/32))^2*exp(pi*i/16)
Respuesta numérica
[src]
z1 = 0.759999153155165 + 0.926061647563445*i
z2 = -1.19222502568095 - 0.117423915896848*i
z3 = 0.926061647563445 - 0.759999153155165*i
z4 = -0.117423915896848 + 1.19222502568095*i
z5 = 0.117423915896848 - 1.19222502568095*i
z6 = -0.926061647563445 + 0.759999153155165*i
z7 = 1.19222502568095 + 0.117423915896848*i
z8 = -0.759999153155165 - 0.926061647563445*i
z8 = -0.759999153155165 - 0.926061647563445*i