ANSWER:
The cosine and sine values are reversed.
[tex]4\cdot cis\mleft(\frac{\pi}{3}\mright)=2+2\sqrt[]{3i}[/tex]
STEP-BY-STEP EXPLANATION:
We can rewrite the function as follows:
[tex]a\cdot cis(\theta)=a\cdot\cos (\theta)+ai\cdot\sin \theta[/tex]
In this case, then it would be:
[tex]\begin{gathered} 4\cdot cis\mleft(\frac{\pi}{3}\mright)=4\cdot\cos \mleft(\frac{\pi}{3}\mright)+4i\cdot\sin \mleft(\frac{\pi}{3}\mright) \\ \cos \mleft(\frac{\pi}{3}\mright)=\frac{1}{2} \\ \sin \mleft(\frac{\pi}{3}\mright)=\frac{\sqrt[]{3}}{2} \\ \text{ replacing} \\ 4\cdot cis\mleft(\frac{\pi}{3}\mright)=4\cdot\frac{1}{2}+4i\cdot\frac{\sqrt[]{3}}{2} \\ 4\cdot cis\mleft(\frac{\pi}{3}\mright)=2+2\sqrt[]{3i} \end{gathered}[/tex]
What it means is that the error was that it was wrong when calculating the cosine and the sine, it has the values inverted
