Given [tex]$z^2=\frac{1}{2}\left[\cos \left(\frac{2 \pi}{5}\right)+i \sin \left(\frac{2 \pi}{5}\right)\right][tex]$[/tex], which lettered point represents [tex]$[/tex]z$[/tex]?

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Answer :

Given the equation \( z^2 = \frac{1}{2} \left[ \cos \left( \frac{2\pi}{5} \right) + i \sin \left( \frac{2\pi}{5} \right) \right] \), we need to determine the point that represents the complex number \( z \).

First, let's rewrite the given complex number on the right-hand side in polar form:
[tex]\[ z^2 = \frac{1}{2} \left[ \cos \left( \frac{2\pi}{5} \right) + i \sin \left( \frac{2\pi}{5} \right) \right]. \][/tex]

1. We start by identifying the magnitude \( r \) and the angle \( \theta \) of the complex number:
- The magnitude of the right-hand side is \( \frac{1}{\sqrt{2}} \), since \( \left| \frac{1}{2} \right| = \frac{1}{2} \) and we take the square root of this magnitude to find \( |z| \).
- The angle \( \theta \) for this complex number is \( \frac{2\pi}{10} = \frac{\pi}{5} \), using the De Moivre's theorem.

2. Next, we find \( z \) by taking the square root of both sides. To do this, we use the fact that:
[tex]\[ z = \sqrt{\frac{1}{\sqrt{2}}} \left[ \cos \left( \frac{\pi}{5} \right) + i \sin \left( \frac{\pi}{5} \right) \right]. \][/tex]

3. Calculate the magnitude of \( z \):
[tex]\[ |z| = \left( \frac{1}{\sqrt{2}} \right)^{1/2} = \left( \frac{1}{2} \right)^{1/4} = \frac{1}{\sqrt[4]{2}}. \][/tex]

4. Determine the real and imaginary parts of \( z \) based on the angle \( \frac{\pi}{5} \):
[tex]\[ z = \frac{1}{\sqrt[4]{2}} \left[ \cos \left( \frac{\pi}{5} \right) + i \sin \left( \frac{\pi}{5} \right) \right]. \][/tex]

5. Plug the angle into the trigonometric functions and find the Cartesian coordinates:
- The real part is \( \frac{1}{\sqrt[4]{2}} \cos \left( \frac{\pi}{5} \right) \approx 0.572 \)
- The imaginary part is \( \frac{1}{\sqrt[4]{2}} \sin \left( \frac{\pi}{5} \right) \approx 0.416 \)

Therefore, the coordinates representing \( z \) are approximately:
[tex]\[ (0.572, 0.416). \][/tex]

So, the lettered point that corresponds to [tex]\( z \)[/tex] on the complex plane is the one with coordinates very close to [tex]\((0.572, 0.416)\)[/tex]. Identify the point that matches or closely matches this result.