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z^6-sqrt3-i=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$\left(z^{6} - \sqrt{3}\right) - i = 0$$
Ya que la potencia en la ecuación es igual a = 6 y miembro libre = i + sqrt(3) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 6 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{6} = \sqrt{3} + i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{6} e^{6 i p} = \sqrt{3} + i$$
donde
$$r = \sqrt[6]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{6 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
es decir
$$\cos{\left(6 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(6 p \right)} = \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{3} + \frac{\pi}{36}$$
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:
$$w_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$w_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$w_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$z_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$z_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
$$\left(z^{6} - \sqrt{3}\right) - i = 0$$
Ya que la potencia en la ecuación es igual a = 6 y miembro libre = i + sqrt(3) complejo,
significa que la ecuación correspondiente no tiene soluciones reales
Las demás 6 raíces son complejas.
hacemos el cambio:
$$w = z$$
entonces la ecuación será así:
$$w^{6} = \sqrt{3} + i$$
Cualquier número complejo se puede presentar que:
$$w = r e^{i p}$$
sustituimos en la ecuación
$$r^{6} e^{6 i p} = \sqrt{3} + i$$
donde
$$r = \sqrt[6]{2}$$
- módulo del número complejo
Sustituyamos r:
$$e^{6 i p} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
Usando la fórmula de Euler hallemos las raíces para p
$$i \sin{\left(6 p \right)} + \cos{\left(6 p \right)} = \frac{\sqrt{3}}{2} + \frac{i}{2}$$
es decir
$$\cos{\left(6 p \right)} = \frac{\sqrt{3}}{2}$$
y
$$\sin{\left(6 p \right)} = \frac{1}{2}$$
entonces
$$p = \frac{\pi N}{3} + \frac{\pi}{36}$$
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:
$$w_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$w_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$w_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$w_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
hacemos cambio inverso
$$w = z$$
$$z = w$$
Entonces la respuesta definitiva es:
$$z_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$z_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
$$z_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2}$$
$$z_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} i \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}}{2}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\ / 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\ /6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\ /6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\
| \/ 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|--|
6 ___ /pi\ 6 ___ /pi\ 6 ___ /pi\ 6 ___ /pi\ | \36/ \36/| \36/ \36/ | \36/ \36/| \36/ \36/ | \36/ \36/| \36/ \36/ | \36/ \36/| \36/ \36/
- \/ 2 *cos|--| - I*\/ 2 *sin|--| + \/ 2 *cos|--| + I*\/ 2 *sin|--| + I*|- ------------- + -------------------| - ------------- - ------------------- + I*|- ------------- - -------------------| - ------------- + ------------------- + I*|------------- + -------------------| + ------------- - ------------------- + I*|------------- - -------------------| + ------------- + -------------------
\36/ \36/ \36/ \36/ \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 2 2
$$\left(\frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) + \left(\left(\left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) + \left(\left(\left(- \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) + \left(\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right)\right) + \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right)\right)\right) + \left(- \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right)\right)$$
=
/6 ___ /pi\ 6 ___ ___ /pi\\ /6 ___ /pi\ 6 ___ ___ /pi\\ / 6 ___ /pi\ 6 ___ ___ /pi\\ / 6 ___ /pi\ 6 ___ ___ /pi\\ |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| |\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| | \36/ \36/| | \36/ \36/| | \36/ \36/| | \36/ \36/| I*|------------- + -------------------| + I*|------------- - -------------------| + I*|- ------------- + -------------------| + I*|- ------------- - -------------------| \ 2 2 / \ 2 2 / \ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right) + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)$$
producto
/ / 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\\ / / 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\\ / /6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\\ / /6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\\
| | \/ 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|--||
/ 6 ___ /pi\ 6 ___ /pi\\ /6 ___ /pi\ 6 ___ /pi\\ | | \36/ \36/| \36/ \36/| | | \36/ \36/| \36/ \36/| | | \36/ \36/| \36/ \36/| | | \36/ \36/| \36/ \36/|
|- \/ 2 *cos|--| - I*\/ 2 *sin|--||*|\/ 2 *cos|--| + I*\/ 2 *sin|--||*|I*|- ------------- + -------------------| - ------------- - -------------------|*|I*|- ------------- - -------------------| - ------------- + -------------------|*|I*|------------- + -------------------| + ------------- - -------------------|*|I*|------------- - -------------------| + ------------- + -------------------|
\ \36/ \36// \ \36/ \36// \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 / \ \ 2 2 / 2 2 /
$$\left(- \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) \left(\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}\right) \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(- \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(- \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)\right) \left(\frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)\right)$$
=
pi*I
2 2 ----
/ /5*pi\ /5*pi\\ / /7*pi\ /7*pi\\ 18
-2*|I*cos|----| + sin|----|| *|- I*cos|----| + sin|----|| *e
\ \ 36 / \ 36 // \ \ 36 / \ 36 //
$$- 2 \left(\sin{\left(\frac{5 \pi}{36} \right)} + i \cos{\left(\frac{5 \pi}{36} \right)}\right)^{2} \left(\sin{\left(\frac{7 \pi}{36} \right)} - i \cos{\left(\frac{7 \pi}{36} \right)}\right)^{2} e^{\frac{i \pi}{18}}$$
-2*(i*cos(5*pi/36) + sin(5*pi/36))^2*(-i*cos(7*pi/36) + sin(7*pi/36))^2*exp(pi*i/18)
Respuesta rápida
[src]
6 ___ /pi\ 6 ___ /pi\
z1 = - \/ 2 *cos|--| - I*\/ 2 *sin|--|
\36/ \36/
$$z_{1} = - \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} - \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
6 ___ /pi\ 6 ___ /pi\
z2 = \/ 2 *cos|--| + I*\/ 2 *sin|--|
\36/ \36/
$$z_{2} = \sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)} + \sqrt[6]{2} i \sin{\left(\frac{\pi}{36} \right)}$$
/ 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\
| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \36/ \36/| \36/ \36/
z3 = I*|- ------------- + -------------------| - ------------- - -------------------
\ 2 2 / 2 2
$$z_{3} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)$$
/ 6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\
| \/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \36/ \36/| \36/ \36/
z4 = I*|- ------------- - -------------------| - ------------- + -------------------
\ 2 2 / 2 2
$$z_{4} = - \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} - \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)$$
/6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\
|\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \36/ \36/| \36/ \36/
z5 = I*|------------- + -------------------| + ------------- - -------------------
\ 2 2 / 2 2
$$z_{5} = - \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(\frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2}\right)$$
/6 ___ /pi\ 6 ___ ___ /pi\\ 6 ___ /pi\ 6 ___ ___ /pi\
|\/ 2 *sin|--| \/ 2 *\/ 3 *cos|--|| \/ 2 *cos|--| \/ 2 *\/ 3 *sin|--|
| \36/ \36/| \36/ \36/
z6 = I*|------------- - -------------------| + ------------- + -------------------
\ 2 2 / 2 2
$$z_{6} = \frac{\sqrt[6]{2} \sqrt{3} \sin{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \cos{\left(\frac{\pi}{36} \right)}}{2} + i \left(- \frac{\sqrt[6]{2} \sqrt{3} \cos{\left(\frac{\pi}{36} \right)}}{2} + \frac{\sqrt[6]{2} \sin{\left(\frac{\pi}{36} \right)}}{2}\right)$$
z6 = 2^(1/6)*sqrt(3)*sin(pi/36)/2 + 2^(1/6)*cos(pi/36)/2 + i*(-2^(1/6)*sqrt(3)*cos(pi/36)/2 + 2^(1/6)*sin(pi/36)/2)
Respuesta numérica
[src]
z1 = -1.118190741335 - 0.0978290135264612*i
z2 = 0.474372959726412 + 1.01729609503589*i
z3 = 0.643817781608586 - 0.919467081509432*i
z4 = -0.474372959726412 - 1.01729609503589*i
z5 = 1.118190741335 + 0.0978290135264612*i
z6 = -0.643817781608586 + 0.919467081509432*i
z6 = -0.643817781608586 + 0.919467081509432*i