Answer:
[tex]z[/tex] is represented by point A.
Step-by-step explanation:
According to the De Moivre's Formula, the power of a complex number of the form [tex]z = r\cdot (\cos \theta + i\,\sin \theta)[/tex] is defined by this formula:
[tex]z^{n} = r^{n}\cdot [\cos n\theta + i\,\sin n\theta][/tex] (1)
Where:
[tex]r[/tex] - Norm of the complex number, dimensionless.
[tex]\theta[/tex] - Direction of the complex number, measured in radians.
[tex]n[/tex] - Power of the resulting complex number, dimensionless.
Given that [tex]z^{2} = \frac{1}{2}\cdot \left[\cos \left(\frac{2\pi}{5} \right)+ i\,\sin \left(\frac{2\pi}{5} \right)\right][/tex], then the following variables are:
[tex]r^{2} = \frac{1}{2}[/tex] (2)
[tex]2\theta = \frac{2\pi}{5}[/tex] (3)
And the norm and direction of the complex number are, respectively:
[tex]r = \frac{\sqrt{2}}{2}[/tex] and [tex]\theta = \frac{\pi}{5}[/tex]
Then, the complex number is:
[tex]z = \frac{\sqrt{2}}{2}\cdot \left[\cos \left(\frac{\pi}{5} \right)+i\,\sin \left(\frac{\pi}{5} \right)\right][/tex]
[tex]z = 0.572 + i\,0.416[/tex]
Which corresponds to point A.
