Problem:
The equation z6+z3+1=0 has one complex root with argument (angle) ΞΈ between 90β and 180β in the complex plane. Determine the degree measure of ΞΈ.
Solution:
Let w=z3. Then the given equation reduces to
w2+w+1=0
whose solutions are 2(β1+i3β)β and 2(β1βi3β)β, with arguments of 120β and 240β, respectively. From these, one finds the following six values for the argument of z:
of these, only z3β and z4β are in the second quadrant. However, since the solutions of z9β1=0 are distinct, and since z3β is a solution of z3β1=0, it cannot be a solution of the original equation. It follows that the desired root is z4β, with degree measure 160β.