Question 1066889
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The complex numbers z,z^4 and z^5 where z = cos(2pi/7)+i sin(2pi/7) are represented by the points P,Q and R respectively in the Argand Diagram. 
If triangle PQR is isosceles, state which sides are equal and it's angles in terms of pi.
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"Argand diagram" is a "scientific name" for the simple classical complex plane with the complex numbers presented by the corresponding points.


<pre>
So, we have the unit circle with the points 

P = z  = cos(2pi/7)+i sin(2pi/7)

Q = {{{z^4}}} = cos(8pi/7)+i sin(8pi/7)

R = {{{z^5}}} = cos(10pi/7)+i sin(10pi/7)

in it.


The arc between the points z and {{{z^4}}} is {{{6pi/7}}} (the difference of arguments of these complex numbers).

The arc between the points z and {{{z^5}}} is again {{{6pi/7}}}.


So, the triangle PQR has congruent sides PQ and PR, since they tighten congruent arcs.
</pre>

Solved.


On complex numbers see the lessons

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