To express [tex]\(\frac{2}{3} \left( \cos 300^\circ + \sin 300^\circ \right)\)[/tex] in the form [tex]\(a + b i\)[/tex], let’s break it down step by step:
1. Identify the angle and its reference:
[tex]\(300^\circ\)[/tex] is in the fourth quadrant, where the cosine function is positive and the sine function is negative.
2. Compute [tex]\(\cos 300^\circ\)[/tex] and [tex]\(\sin 300^\circ\)[/tex]:
Standard values for these trigonometric functions:
[tex]\[
\cos 300^\circ = \frac{1}{2}
\][/tex]
[tex]\[
\sin 300^\circ = -\frac{\sqrt{3}}{2}
\][/tex]
3. Substitute the values into the expression:
[tex]\[
\frac{2}{3} \left( \cos 300^\circ + \sin 300^\circ \right) = \frac{2}{3} \left( \frac{1}{2} + \left( -\frac{\sqrt{3}}{2} \right) \right)
\][/tex]
4. Simplify the expression inside the parentheses:
[tex]\[
\frac{1}{2} + \left( -\frac{\sqrt{3}}{2} \right) = \frac{1}{2} - \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2} + \frac{1}{2}
\][/tex]
5. Multiply by [tex]\(\frac{2}{3}\)[/tex]:
Distribute [tex]\(\frac{2}{3}\)[/tex] across the expression:
[tex]\[
\frac{2}{3} \cdot \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right) = \frac{2}{3} \cdot \frac{1}{2} + \frac{2}{3} \cdot \left(-\frac{\sqrt{3}}{2}\right)
\][/tex]
[tex]\[
= \frac{2}{6} + \left( -\frac{2\sqrt{3}}{6} \right)
\][/tex]
[tex]\[
= \frac{1}{3} - \frac{\sqrt{3}}{3}
\][/tex]
6. Identify the standard form:
Comparing the final simplified result [tex]\(\frac{1}{3} - \frac{\sqrt{3}}{3}\)[/tex] with the given options, we see that it matches:
[tex]\[
\frac{1}{3} - \frac{\sqrt{3}}{3}
\][/tex]
Therefore, the correct expression is:
[tex]\[ \boxed{\frac{1}{3} - \frac{\sqrt{3}}{3}} \][/tex]
