To determine the argument of the complex number [tex]\( -1 + \sqrt{3}i \)[/tex], we'll follow these steps:
1. Identify the components of the complex number:
- The real part [tex]\(a\)[/tex] is [tex]\(-1\)[/tex].
- The imaginary part [tex]\(b\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
2. Calculate the argument [tex]\(\theta\)[/tex] using the arctangent function:
[tex]\[
\theta = \text{atan2}(b, a)
\][/tex]
Here, [tex]\(\text{atan2}(b, a)\)[/tex] provides the angle [tex]\(\theta\)[/tex] in radians such that:
[tex]\[
\theta = \text{atan2}(\sqrt{3}, -1)
\][/tex]
3. Convert the argument from radians to degrees:
[tex]\[
\theta \approx 1.0471975511965976 \text{ radians}
\][/tex]
To convert this into degrees, we use the conversion factor [tex]\(\frac{180}{\pi}\)[/tex]:
[tex]\[
\theta_{\text{degrees}} = \theta \times \frac{180}{\pi} \approx 1.0471975511965976 \times \frac{180}{\pi} \approx 60^\circ
\][/tex]
4. Round to the nearest integer degree if necessary:
[tex]\[
\theta_{\text{rounded degrees}} \approx 60^\circ
\][/tex]
Therefore, the argument of the complex number [tex]\(-1 + \sqrt{3}i\)[/tex] is [tex]\(\boxed{60^\circ}\)[/tex].
