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#1 2013-06-02 00:32:17

fm_hlp
Guest

complex number

By writing the equation


in the form

show that its roots are


please provide a step to step answer to teach me
i cant seem to solve

#2 2013-06-02 03:20:00

SteveB
Member
Registered: 2013-03-07
Posts: 547

Re: complex number

I am not sure about this because I cannot do the very last bit of trigonometry but this is what I have got so far:

Using a right angled triangle in the Argand diagram:




Use the rule of multiplying by the complex conjugate pair:



Using standard trigonometric results we need this in terms of cot(A/2):








I have put in the minus sign in now. Well spotted anonimnystefy.
k = 0  is not a solution I agree Stefy, but why does my answer not work for k = 0 ?

EDIT: I now can see that cot(0) is not possible. Given that cot X = 1/(tan X) if X=0 then we have a division by zero.
Therefore the argument is not valid for this case.

Last edited by SteveB (2013-06-02 18:28:32)

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#3 2013-06-02 04:17:50

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,860

Re: complex number

Hi SteveB

Have you tried using the half-angle formulas for sine and cosine? (the answer provided has k*pi/5, not 2*k*pi/5 smile)


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#4 2013-06-02 04:44:51

SteveB
Member
Registered: 2013-03-07
Posts: 547

Re: complex number

I was working on those formulas as you were writing that post.
I think it works and produces the right result.
There are 5 solutions.

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#5 2013-06-02 04:55:14

anonimnystefy
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From: The Foundation
Registered: 2011-05-23
Posts: 14,860

Re: complex number

In this steps you lose s minus sign:


There are 4 solutions. k=0 is not a solution.

Last edited by anonimnystefy (2013-06-02 04:59:15)


“Here lies the reader who will never open this book. He is forever dead.

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#6 2013-06-02 05:00:46

SteveB
Member
Registered: 2013-03-07
Posts: 547

Re: complex number

Yes. I agree. If you let k=0 then z = 1
Obviously (1 + 1)/1 = 2
So this cannot give us 1 when raised to the power of 5.
Why does my algebraic argument not work for k = 0
Usually a fifth root gives us five solutions. (????)

I can see your point about the negative sign as well .....

Last edited by SteveB (2013-06-02 05:03:40)

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#7 2013-06-02 05:03:50

fm_hlp
Guest

Re: complex number

wow
thanks a lot

#8 2013-06-02 05:04:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,860

Re: complex number

Well, actually, when k=0, z is not defined (cot(0)=not defined). But, the first troubles with k=0 begin when we take the reciprocal of both sides of "1/z=..." .

Last edited by anonimnystefy (2013-06-02 05:05:28)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2013-06-02 05:10:48

SteveB
Member
Registered: 2013-03-07
Posts: 547

Re: complex number

Thanks anonimnystefy. I thought there might be a division by zero somewhere !!

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#10 2013-06-02 05:12:37

fm_hlp
Guest

Re: complex number


its a polynomial of power 4 as it simplifies to

#11 2013-06-02 05:29:14

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,860

Re: complex number

fm_hlp wrote:


its a polynomial of power 4 as it simplifies to

That is not correct.

SteveB wrote:

Thanks anonimnystefy. I thought there might be a division by zero somewhere !!

You're welcome!

Last edited by anonimnystefy (2013-06-02 05:29:43)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#12 2013-06-02 05:38:19

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,259

Re: complex number

hi Stefy

Why isn't it correct?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#13 2013-06-02 05:40:49

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 14,860

Re: complex number

Oh, sorry. I didn't understand what he wanted to say. And, anyway, using that would be very hard for a hand method.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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