To find [tex]\( z^3 \)[/tex] for the given complex number [tex]\( z = 4 \left[\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right] \)[/tex], we will use the properties of complex numbers in polar form.
Given:
[tex]\[ z = 4 \left[\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right] \][/tex]
First, let's recall the exponential form of a complex number:
[tex]\[ z = r \text{cis}(\theta) \][/tex]
where [tex]\( \text{cis}(\theta) = \cos(\theta) + i \sin(\theta) \)[/tex].
Here, [tex]\( r = 4 \)[/tex] and [tex]\( \theta = \frac{\pi}{2} \)[/tex].
Using De Moivre's Theorem, which states:
[tex]\[ \left( r \text{cis}(\theta) \right)^n = r^n \text{cis}(n\theta) \][/tex]
We want to find [tex]\( z^3 \)[/tex]:
[tex]\[ z^3 = \left[ 4 \text{cis}\left( \frac{\pi}{2} \right) \right]^3 = 4^3 \text{cis}\left( 3 \cdot \frac{\pi}{2} \right) \][/tex]
Calculating the magnitude:
[tex]\[ 4^3 = 64 \][/tex]
Calculating the angle:
[tex]\[ 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \][/tex]
Thus:
[tex]\[ z^3 = 64 \left[\cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right)\right] \][/tex]
So, the correct answer is:
[tex]\[ 64 \left[\cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right)\right] \][/tex]
