SOLUTION: 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

Question 1066889: 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.
Answer by ikleyn(52865) (Show Source): You can put this solution on YOUR website!
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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.

So, we have the unit circle with the points 

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

Q =  = cos(8pi/7)+i sin(8pi/7)

R =  = cos(10pi/7)+i sin(10pi/7)

in it.


The arc between the points z and  is  (the difference of arguments of these complex numbers).

The arc between the points z and  is again .


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

Solved.

On complex numbers see the lessons
    - Complex numbers and arithmetic operations on them
    - Complex plane
    - Addition and subtraction of complex numbers in complex plane
    - Multiplication and division of complex numbers in complex plane
    - Raising a complex number to an integer power
in this site.

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

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

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