Answer:
The complex number a+ib is;
[tex]1+i\sqrt[]{3}[/tex]Explanation:
Given the complex number;
[tex]a+ib[/tex]The modulus of the complex number is given as;
[tex]\begin{gathered} \sqrt[]{a^2+b^2}=2 \\ a^2+b^2=2^2 \\ a^2+b^2=4 \\ a^2=4-b^2\text{ -------- 1} \end{gathered}[/tex]Also the argument of the complex number;
[tex]\begin{gathered} arg(z)=\tan ^{-1}(\frac{b}{a})=\frac{\pi}{3} \\ \frac{b}{a}=\tan (\frac{\pi}{3}) \\ \frac{b}{a}=\sqrt[]{3} \\ b=a\sqrt[]{3}--------2 \end{gathered}[/tex]substituting equation 2 into equation 1;
[tex]\begin{gathered} a^2=4-b^2\text{ } \\ a^2=4-(a\sqrt[]{3})^2\text{ } \\ a^2=4-a^2(3) \\ a^2+3a^2=4 \\ 4a^2=4 \\ a^2=\frac{4}{4} \\ a^2=1 \\ a=1 \end{gathered}[/tex]Substituting a=1 into equation 2;
[tex]\begin{gathered} b=a\sqrt[]{3} \\ b=1\times\sqrt[]{3} \\ b=\sqrt[]{3} \end{gathered}[/tex]Therefore, the complex number a+ib is;
[tex]1+i\sqrt[]{3}[/tex]