Question

Find [5(cos 330 degrees + I sin 330 degrees)]^3

Answer 1

Given a complex number in the form:

`z= \rho [\cos \theta + i \sin \theta]`
The nth-power of this number,
`z^n`, can be calculated as follows:

- the modulus of
`z^n` is equal to the nth-power of the modulus of z, while the angle of
`z^n` is equal to n multiplied the angle of z, so:

`z^n = \rho^n [\cos n\theta + i \sin n\theta ]`
In our case, n=3, so
`z^3` is equal to

`z^3 = \rho^3 [\cos 3 \theta + i \sin 3 \theta ] = (5^3) [\cos (3 \cdot 330^(\circ)) + i \sin (3 \cdot 330^(\circ)) ]` (1)
And since

`3 \cdot 330^(\circ) = 990^(\circ) = 2\pi +270^(\circ)`
and both sine and cosine are periodic in
`2 \pi`, (1) becomes

`z^3 = 125 [\cos 270^(\circ) + i \sin 270^(\circ) ]`