Escribir las siguientes ecuaciones de la forma:
![\mathbb{Z}=\mathbb{R}\left \{ A\\e ^{j\left ( wt+\Theta j \right )} \right \}](https://latex.codecogs.com/gif.latex?\mathbb{Z}=\mathbb{R}\left&space;\{&space;A\\e&space;^{j\left&space;(&space;wt+\Theta&space;j&space;\right&space;)}&space;\right&space;\})
1) Resolver:
![G1= \cos \left ( wt+\frac{\pi }{4} \right )- 2 \sin \left ( wt+\frac{\pi }{6} \right )](https://latex.codecogs.com/gif.latex?G1=&space;\cos&space;\left&space;(&space;wt+\frac{\pi&space;}{4}&space;\right&space;)-&space;2&space;\sin&space;\left&space;(&space;wt+\frac{\pi&space;}{6}&space;\right&space;))
sabemos que:
llevamos al seno a la expresión coseno
![G1= \cos \left ( wt+\frac{\pi }{4} \right ) + 2 \cos \left ( wt+\frac{\pi}{6}+\frac{\pi}{2} \right )](https://latex.codecogs.com/gif.latex?G1=&space;\cos&space;\left&space;(&space;wt+\frac{\pi&space;}{4}&space;\right&space;)&space;+&space;2&space;\cos&space;\left&space;(&space;wt+\frac{\pi}{6}+\frac{\pi}{2}&space;\right&space;))
![G1= \cos \left ( wt+\frac{\pi }{4} \right ) + 2 \cos \left ( wt+\frac{2\pi}{3} \right )](https://latex.codecogs.com/gif.latex?G1=&space;\cos&space;\left&space;(&space;wt+\frac{\pi&space;}{4}&space;\right&space;)&space;+&space;2&space;\cos&space;\left&space;(&space;wt+\frac{2\pi}{3}&space;\right&space;))
Ahora a la forma compleja:
![G1= e^{wt}e^{j\frac{\pi}{4}}+2e^{wt} e^{j2\frac{2\pi}{3}}](https://latex.codecogs.com/gif.latex?G1=&space;e^{wt}e^{j\frac{\pi}{4}}+2e^{wt}&space;e^{j2\frac{2\pi}{3}})
Entonces:
![G1=\left ( \cos \left ( wt \right ) + j\sin \left ( wt \right )\right )\left [ \cos \left ( \frac{\pi}{4} \right )+j\sin \left ( \frac{\pi}{4} \right ) +2\cos \left ( \frac{2\pi}{3} \right )+2j\sin \left ( \frac{2\pi}{3} \right )\right ]](https://latex.codecogs.com/gif.latex?G1=\left&space;(&space;\cos&space;\left&space;(&space;wt&space;\right&space;)&space;+&space;j\sin&space;\left&space;(&space;wt&space;\right&space;)\right&space;)\left&space;[&space;\cos&space;\left&space;(&space;\frac{\pi}{4}&space;\right&space;)+j\sin&space;\left&space;(&space;\frac{\pi}{4}&space;\right&space;)&space;+2\cos&space;\left&space;(&space;\frac{2\pi}{3}&space;\right&space;)+2j\sin&space;\left&space;(&space;\frac{2\pi}{3}&space;\right&space;)\right&space;])
![G1= \left ( \cos \left ( wt \right ) + j\sin \left ( wt \right )\right )\left [ \frac{\sqrt{2}}{2}+\frac{j\sqrt{2}}{2}+2\left ( \frac{-1}{2} \right )+2j\left ( \frac{\sqrt{3}}{2} \right )\right ]](https://latex.codecogs.com/gif.latex?G1=&space;\left&space;(&space;\cos&space;\left&space;(&space;wt&space;\right&space;)&space;+&space;j\sin&space;\left&space;(&space;wt&space;\right&space;)\right&space;)\left&space;[&space;\frac{\sqrt{2}}{2}+\frac{j\sqrt{2}}{2}+2\left&space;(&space;\frac{-1}{2}&space;\right&space;)+2j\left&space;(&space;\frac{\sqrt{3}}{2}&space;\right&space;)\right&space;])
![G1= \left ( \cos \left ( wt \right ) +j \sin \left ( wt \right )\right )\left [ \frac{\sqrt{2}}{2}-1+j\left ( \frac{\sqrt{2}}{2} + \sqrt{3}\right ) \right ]](https://latex.codecogs.com/gif.latex?G1=&space;\left&space;(&space;\cos&space;\left&space;(&space;wt&space;\right&space;)&space;+j&space;\sin&space;\left&space;(&space;wt&space;\right&space;)\right&space;)\left&space;[&space;\frac{\sqrt{2}}{2}-1+j\left&space;(&space;\frac{\sqrt{2}}{2}&space;+&space;\sqrt{3}\right&space;)&space;\right&space;])
![G1\approx \left ( \cos \left ( wt \right ) +j\sin \left ( wt \right )\right )\left [ -0.29+j2.43 \right ]](https://latex.codecogs.com/gif.latex?G1\approx&space;\left&space;(&space;\cos&space;\left&space;(&space;wt&space;\right&space;)&space;+j\sin&space;\left&space;(&space;wt&space;\right&space;)\right&space;)\left&space;[&space;-0.29+j2.43&space;\right&space;])
Para graficar solo nos interesa los valores numéricos.
Es recomendable graficar el punto
![A=\sqrt{\left ( \frac{\sqrt{2}}{2}-1\right )^{2}+\left ( \frac{\sqrt{2}}{2}+\sqrt{3} \right )^{2}}](https://latex.codecogs.com/gif.latex?A=\sqrt{\left&space;(&space;\frac{\sqrt{2}}{2}-1\right&space;)^{2}+\left&space;(&space;\frac{\sqrt{2}}{2}+\sqrt{3}&space;\right&space;)^{2}})
![A= 2.45](https://latex.codecogs.com/gif.latex?A=&space;2.45)
![A= 2.45](https://latex.codecogs.com/gif.latex?A=&space;2.45)
![\theta = \tanh^{-1}\left ( \frac{\mathbb{I}}{\mathbb{Re}} \right )](https://latex.codecogs.com/gif.latex?\theta&space;=&space;\tanh^{-1}\left&space;(&space;\frac{\mathbb{I}}{\mathbb{Re}}&space;\right&space;))
![\theta =\tan^{-1}\left ( \frac{2.43}{-0.29} \right )](https://latex.codecogs.com/gif.latex?\theta&space;=\tan^{-1}\left&space;(&space;\frac{2.43}{-0.29}&space;\right&space;))
![\theta =-83.19441956^{\circ }](https://latex.codecogs.com/gif.latex?\theta&space;=-83.19441956^{\circ&space;})
El ángulo no es principal por eso le sumamos 180 grados
![\theta =96.80558044^{\circ }](https://latex.codecogs.com/gif.latex?\theta&space;=96.80558044^{\circ&space;})
Finalmente:
![Z=2.45\left \{ e^{j96.80558044^{\circ }} \right \}](https://latex.codecogs.com/gif.latex?Z=2.45\left&space;\{&space;e^{j96.80558044^{\circ&space;}}&space;\right&space;\})
Nota:
![e^{jwt} = \left ( \cos \left ( wt \right ) + j\sin \left ( wt \right )\right )](https://latex.codecogs.com/gif.latex?e^{jwt}&space;=&space;\left&space;(&space;\cos&space;\left&space;(&space;wt&space;\right&space;)&space;+&space;j\sin&space;\left&space;(&space;wt&space;\right&space;)\right&space;))
su módulo es igual a 1 y el ángulo a 0 por ende no afecta.
1) Resolver:
sabemos que:
llevamos al seno a la expresión coseno
Ahora a la forma compleja:
Entonces:
Para graficar solo nos interesa los valores numéricos.
Es recomendable graficar el punto
El ángulo no es principal por eso le sumamos 180 grados
Finalmente:
Nota:
su módulo es igual a 1 y el ángulo a 0 por ende no afecta.
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