[tex]\bf \stackrel{a}{12}i+\stackrel{b}{4\sqrt{3}}j\implies \ \textless \ 12~,~4\sqrt{3}\ \textgreater \ \qquad
\begin{cases}
\stackrel{magnitude}{r}=\sqrt{a^2+b^2}\\\\
\theta =tan^{-1}\left( \frac{b}{a} \right)
\end{cases}\\\\
-------------------------------\\\\
r=\sqrt{12^2+(4\sqrt{3})^2}\implies r=\sqrt{144+(4^2\sqrt{3^2})}[/tex]
[tex]\bf r=\sqrt{144+(16\cdot 3)}\implies r=\sqrt{144+48}\implies r=\sqrt{192}
\\\\\\
r=\sqrt{64\cdot 3}\implies r=\sqrt{8^2\cdot 3}\implies r=8\sqrt{3}\\\\
-------------------------------\\\\
\theta =tan^{-1}\left( \cfrac{4\sqrt{3}}{12} \right)\implies \theta =tan^{-1}\left( \cfrac{\sqrt{3}}{3} \right)\implies \theta =
\begin{cases}
\frac{\pi }{6}\leftarrow \\\\
\frac{7\pi }{6}
\end{cases}[/tex]
notice, those two angles have a valid tangent value, however, notice the "a" and "b" components, their signs are +a and +b, meaning the angle is in the first quadrant, thus, it has to be angle in the first quadrant then.
