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Trigonometric Complex Numbers

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1. Homework Statement
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how are the solutions of the fourth roots pi/2? how do you get pi/2, you know the thing after "cos" and "isin" ?
 
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Answers and Replies

1,631
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First write it in polar form, or trigonometric form, however you call it, and after that use de moivre's formula to find its roots.

let z be a complex nr.

z=a+bi, writing it in polar forms : [tex]a=r cos(\theta),b=\ro sin\theta[/tex]


So,

[tex] z=r (cos\theta+isin\theta)[/tex]

now

[tex]z^{\frac{1}{n}}=r^{\frac{1}{n}}(cos\frac{\theta +2k\pi}{n}+isin{\frac{\theta+2k\pi}{n})[/tex]

Now all you need to do is figure out what [tex]\theta [/tex] is, and your fine.

Or if you want the exponential representation of a complex nr:

[tex]e^{ix}=cosx+isinx[/tex]

[tex]e^{i\frac{\pi}{2}}=i[/tex] so we get

[tex] i=cos\frac{\pi}{2}+isin\frac{\pi}{2}[/tex]
 
Last edited:
19
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[tex]e^{i\frac{\pi}{2}}=i[/tex]
how do you know that x = pi/2
 
1,631
4
well z=i, is a complex nr right. Following my elaboration above we have a=0, b=1, right?

so [tex]\theta =\arctan\frac{1}{0}-->\frac{\pi}{2}[/tex] loosly speaking.

so the exponential form of i is what i wrote [tex] i=e^{i\frac{\pi}{2}}[/tex]

SO to find the four roots of i just follow de moivres formula that i wrote above, letting k=0,1,2,3.
 
19
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how do you know a=0 and b=1? isnt it a=1 and b=-1 or did i do something wrong
 
HallsofIvy
Science Advisor
Homework Helper
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Your number is i. If you write that in the form a+ bi, what are a and b?
 

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