To determine which of the given options is a square root of the complex number [tex]\(3\left[\cos \left(\frac{4 \pi}{9}\right)+i \sin \left(\frac{4 \pi}{9}\right)\right]\)[/tex], we can follow these steps:
### Step-by-Step Solution
Step 1: Convert the given complex number into its polar form.
Given:
[tex]\[ 3\left[\cos \left(\frac{4\pi}{9}\right) + i \sin \left(\frac{4\pi}{9}\right)\right] \][/tex]
This can be written in polar form as:
[tex]\[ 3 \text{cis} \left(\frac{4\pi}{9}\right) \][/tex]
where [tex]\(\text{cis}(\theta) = \cos(\theta) + i \sin(\theta)\)[/tex].
Step 2: Find the magnitude of the square root.
The magnitude of the original complex number is 3. The magnitude of the square root will be the square root of 3:
[tex]\[ \sqrt{3} \][/tex]
Step 3: Determine the arguments for the possible square roots.
The arguments (angles) of the square roots will be:
[tex]\[ \theta_1 = \frac{1}{2} \left( \frac{4\pi}{9} + 2k\pi \right) \][/tex]
where [tex]\( k \)[/tex] is an integer. Since a complex number has two square roots, we consider [tex]\( k = 0 \)[/tex] and [tex]\( k = 1 \)[/tex].
For [tex]\( k = 0 \)[/tex]:
[tex]\[ \theta_1 = \frac{1}{2} \left( \frac{4\pi}{9} \right) = \frac{2\pi}{9} \][/tex]
For [tex]\( k = 1 \)[/tex]:
[tex]\[ \theta_2 = \frac{1}{2} \left( \frac{4\pi}{9} + 2\pi \right) = \frac{1}{2} \left( \frac{4\pi}{9} + \frac{18\pi}{9} \right) = \frac{1}{2} \left(\frac{22\pi}{9}\right) = \frac{11\pi}{9} \][/tex]
Step 4: Write the possible square roots in polar form.
The two possible square roots are:
[tex]\[ \sqrt{3} \text{cis}\left(\frac{2\pi}{9}\right) \][/tex]
and
[tex]\[ \sqrt{3} \text{cis}\left(\frac{11\pi}{9}\right) \][/tex]
Step 5: Match the options with the calculated roots.
We need to compare the above possible square roots with the given choices.
1. [tex]\(\sqrt{3}\left[\cos \left(\frac{4 \pi}{45}\right)+i \sin \left(\frac{4 \pi}{45}\right)\right]\)[/tex]
2. [tex]\(\sqrt{3}\left[\cos \left(\frac{\pi}{9}\right)+i \sin \left(\frac{\pi}{9}\right)\right]\)[/tex]
3. [tex]\(\sqrt{3}\left[\cos \left(\frac{49 \pi}{60}\right)+i \sin \left(\frac{49 \pi}{60}\right)\right]\)[/tex]
4. [tex]\(\sqrt{3}\left[\cos \left(\frac{11 \pi}{9}\right)+i \sin \left(\frac{11 \pi}{9}\right)\right]\)[/tex]
From these calculations, it's clear that the second possible square root, [tex]\(\sqrt{3} \text{cis}\left(\frac{11\pi}{9}\right)\)[/tex], matches one of the given choices:
[tex]\[ \sqrt{3}\left[\cos \left(\frac{11 \pi}{9}\right)+i \sin \left(\frac{11 \pi}{9}\right)\right] \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\sqrt{3}\left[\cos \left(\frac{11 \pi}{9}\right)+i \sin \left(\frac{11 \pi}{9}\right)\right]} \][/tex]
This corresponds to option 4.
