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
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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(52858) (Show Source):
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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.
