Tuesday, 21 April 2015

absolute value - Complex number with z to the power of 4



I have to find all zC for which BOTH of the following is true:



1) |z|=1



2) |z4+1|=1



I understand that the 1) is a unit circle, but I can't find out what would be the 2).

Calculating the power of (x+yi) seems wrong because I get a function with square root of polynomial of the 8th degree.



I've even tried to equate the two, since their modulo is the same, like this:



|z|=|z4+1|



and then separating two possibilities in which the equation holds, but I get only



z4±z+1=0




which I don't know how to calculate.



I guess there is an elegant solution which I am unable to see...



Any help?


Answer



It seems helpful to square both sides. From the second equation, you get 1=(z4+1)(¯z4+1)=(z¯z)4+z4+¯z4+1=z4+¯z4+2. That is, z4+¯z4=1.



The first equation lets you write z=eiθ, so you have 1=e4iθ+e4iθ=2cos(4θ); in other words, cos(4θ)=12. The solutions of this are well-known.


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