Question

If \( z = 2(\cos 60^\circ + i \sin 60^\circ) \), find the value of \( z^3 \).

• A. -8
• B. -10
• C. -2
• D. -22

Hint:

To compute powers of complex numbers in trigonometric form, use De Moivre's theorem. Verify carefully by converting to rectangular form if the result seems inconsistent with options.

Answer 1

The Correct Option is A

Given: \[ z = 2(\cos 60^\circ + i \sin 60^\circ) = 1 + i \sqrt{3} \] Using binomial expansion: \[ z^3 = (1 + i \sqrt{3})^3 \] First, \[ z^2 = (1 + i \sqrt{3})^2 = 1 + 2 i \sqrt{3} + (i \sqrt{3})^2 = 1 + 2 i \sqrt{3} - 3 = -2 + 2 i \sqrt{3} \] Then, \[ z^3 = z^2 \times z = (-2 + 2 i \sqrt{3})(1 + i \sqrt{3}) \] \[ = -2 - 2 i \sqrt{3} + 2 i \sqrt{3} + 2 (i \sqrt{3})^2 = -2 + 0 + 2(-3) = -2 - 6 = -8 \] Hence, \[ z^3 = -8 \]