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This problem trains for: AMC-12, AIME.
The complex number:
is represented in the complex plane by the point P.
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:
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:
Similarly, if the number z2 rotates the number z by 150° clockwise, then it must have the trigonometric form: