The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: AMC-12, AIME.

The complex number:

equation

is represented in the complex plane by the point P.

figure

The product between this number and a number z1 is represented by the point R.

The product between this number and a number z2 is represented by the point T.

What is the product:

equation

Since the complex number z has absolute value 1 and the products are on the same circle, then the products also have absolute value 1, and so must the numbers z1 and z2 have.

If the number z1 rotates the number z by 120° counter-clockwise, then it must have the trigonometric form:

equation

Similarly, if the number z2 rotates the number z by 150° clockwise, then it must have the trigonometric form:

equation

and

equation