Question 1152837
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<pre>
Using trigonometric presentation of complex numbers (same as polar presentation),

I may assume that {{{z[1]}}} = {{{r*cis(phi)}}},

where "r" is the modulus and {{{phi}}} is the argument, i.e. the polar angle of the number {{{z[1]}}}.


Here I use the standard notation {{{cis(phi)}}} = {{{cos(phi) + i*sin(phi)}}} for complex numbers.


From the condition,  {{{z[2]}}} = {{{r*cis(phi+60^o)}}}  with the same modulus "r" and the argument {{{phi+60^o)}}, i/e. {{{phi}}} rotated by 60°.


Since the formula  {{{(z[1])^2+(z[2])^2}}} = {{{z[1]*z[2]}}} is uniform of the degree 2, the modules will cancel each other in both sides,

so I can forget about them and consider my complex numbers {{{z[1]}}} and {{{z[2]}}} as the unit vectors beginning at 0 (zero point).


So, I need to prove that 


    {{{cis^2(phi)}}} + {{{cis^2(phi+60^o)}}} = {{{cis(phi)*cis(phi+60^0)}}}.


You remember that when complex numbers are multiplied, their arguments added.


Therefore,  {{{cis^2(phi)}}} = {{{cis(2*phi)}}};  {{{cis^2(phi+60^o)}}} = {{{cis(2*phi+120^o)}}}.


In other words, the angle between unit vectors {{{cis^2(phi)}}} and {{{cis^2(phi+60^o)}}} is 120°.


So, {{{cis^2(phi)}}} and {{{cis^2(phi+60^o)}}} are the sides of the rhombus with the degree of 120° between them.


According to the "parallelogram rule" of adding complex numbers (and vectors),

the sum of the numbers {{{cis^2(phi)}}} and {{{cis^2(phi+60^o)}}} is the SHORT DIAGONAL of this rhombus,

i.e. the unit vector {{{cis(2*phi+60^o)}}}.


Thus I proved that  {{{cis^2(phi)}}} + {{{cis^2(phi+60^o)}}} = {{{cis(2*phi+60^o)}}}.     (1)


From the other side,  {{{z[1]*z[2]}}} = {{{cis(phi)*cis(phi+60^o)}}} = {{{cis(2*phi+60^o)}}}, 

i.e. the same complex number as (1).


Thus I proved that   {{{(z[1])^2}}} + {{{(Z[2])^2}}} = {{{z[1]*z[2]}}}.
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;C O M P L E T E D &nbsp;&nbsp;and  &nbsp;&nbsp;S O L V E D.



On complex numbers, see introductory lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-numbers-and-arithmetical-operations.lesson>Complex numbers and arithmetical operations on them</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Complex-plane.lesson>Complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Addition-and-subtraction-of-complex-numbers-in-complex-plane.lesson>Addition and subtraction of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Multiplication-and-division-of-complex-numbers-in-complex-plane-.lesson>Multiplication and division of complex numbers in complex plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/complex/Raising-a-complex-number-to-an-integer-power.lesson>Raising a complex number to an integer power</A>

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Complex numbers</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.