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(A) The vertices of an equilateral triangle inscribed in a circle with center at the origin are given by rotating the first vertex by $120^\circ$ ($2\

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$ -2, 1 - i\sqrt{3} $

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(A) The vertices of an equilateral triangle inscribed in a circle with center at the origin are given by rotating the first vertex by $120^\circ$ ($2\pi/3$ radians) and $240^\circ$ ($4\pi/3$ radians) about the origin.Given $z_1 = 1 + i\sqrt{3} = 2e^{i\pi/3}$.The other vertices are $z_2 = z_1 e^{i2\pi/3} = 2e^{i\pi/3} e^{i2\pi/3} = 2e^{i\pi} = -2$.And $z_3 = z_1 e^{i4\pi/3} = 2e^{i\pi/3} e^{i4\pi/3} = 2e^{i5\pi/3} = 2(\cos(5\pi/3) + i\sin(5\pi/3)) = 2(1/2 - i\sqrt{3}/2) = 1 - i\sqrt{3}$.Thus,the values of $z_3$ and $z_2$ are $1 - i\sqrt{3}$ and $-2$ respectively,or vice versa depending on the order of rotation. Comparing with the options,option $A$ provides the correct set of values.

Suppose $z_1, z_2, z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z| = 2$. If $z_1 = 1 + i\sqrt{3}$,then the values of $z_3$ and $z_2$ are respectively:

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