Given a complex number in the form:
[tex]z= \rho [\cos \theta + i \sin \theta][/tex]
The nth-power of this number, [tex]z^n[/tex], can be calculated as follows:
- the modulus of [tex]z^n[/tex] is equal to the nth-power of the modulus of z, while the angle of [tex]z^n[/tex] is equal to n multiplied the angle of z, so:
[tex]z^n = \rho^n [\cos n\theta + i \sin n\theta ][/tex]
In our case, n=3, so [tex]z^3[/tex] is equal to
[tex]z^3 = \rho^3 [\cos 3 \theta + i \sin 3 \theta ] = (5^3) [\cos (3 \cdot 330^{\circ}) + i \sin (3 \cdot 330^{\circ}) ][/tex] (1)
And since
[tex]3 \cdot 330^{\circ} = 990^{\circ} = 2\pi +270^{\circ}[/tex]
and both sine and cosine are periodic in [tex]2 \pi[/tex], (1) becomes
[tex]z^3 = 125 [\cos 270^{\circ} + i \sin 270^{\circ} ][/tex]
