To find the argument of the complex number [tex]\(\frac{2+6 \sqrt{3} i}{5+\sqrt{3} i}\)[/tex], we can proceed through the following steps:
1. Express the Complex Numbers:
- Let [tex]\( z_1 = 2 + 6\sqrt{3} i \)[/tex].
- Let [tex]\( z_2 = 5 + \sqrt{3} i \)[/tex].
2. Perform the Division:
We need to divide [tex]\( z_1 \)[/tex] by [tex]\( z_2 \)[/tex]:
[tex]\[
\frac{z_1}{z_2} = \frac{2 + 6\sqrt{3} i}{5 + \sqrt{3} i}
\][/tex]
3. Calculate the Argument:
The argument (or phase) of a complex number [tex]\( z = a + bi \)[/tex] is given by [tex]\( \operatorname{Arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \)[/tex].
However, for our purposes, it's convenient to use advanced calculation tools, but the outcome gives us a specific argument value for the resulting complex number.
Through this process, we remember that the argument we find needs to be compared to possible choices given in terms of radians.
4. Find the Closest Choice:
Given our choices in radians:
- [tex]\( \frac{\pi}{3} \)[/tex]
- [tex]\( \frac{\pi}{4} \)[/tex]
- [tex]\( \frac{2 \pi}{3} \)[/tex]
- [tex]\( \pi \)[/tex]
Upon calculation, the argument of the resulting complex number from the division is approximately [tex]\(1.0471975511965979\)[/tex] radians. We now need to identify the nearest value from the given choices.
5. Compare with Choices:
- [tex]\(\frac{\pi}{3} \approx 1.0471975511965976 \)[/tex]
- [tex]\(\frac{\pi}{4} \approx 0.7853981633974483 \)[/tex]
- [tex]\(\frac{2\pi}{3} \approx 2.0943951023931953 \)[/tex]
- [tex]\( \pi \approx 3.141592653589793 \)[/tex]
Clearly, the value [tex]\(1.0471975511965979\)[/tex] is closest to [tex]\(\frac{\pi}{3}\)[/tex].
Thus, the argument of the complex number [tex]\(\frac{2+6 \sqrt{3} i}{5+\sqrt{3} i}\)[/tex] is:
[tex]\[
\boxed{\frac{\pi}{3}}
\][/tex]
