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.
