We begin by evaluating the trigonometric expressions.

[latex]\begin{gathered}\cos \left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}\\\sin \left(\frac{\pi }{6}\right)=\frac{1}{2}\end{gathered}[/latex]

After substitution, the complex number is

[latex]z=12\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)[/latex]

We apply the distributive property:

[latex]\begin{align}z&=12\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right) \\ &=\left(12\right)\frac{\sqrt{3}}{2}+\left(12\right)\frac{1}{2}i \\ &=6\sqrt{3}+6i \end{align}[/latex]

The rectangular form of the given point in complex form is [latex]6\sqrt{3}+6i[/latex].

If [latex]\tan \theta =\frac{5}{12}[/latex], and [latex]\tan \theta =\frac{y}{x}[/latex], we first determine [latex]r=\sqrt{{x}^{2}+{y}^{2}}=\sqrt{{12}^{2}+{5}^{2}}=13\text{.}[/latex] We then find [latex]\cos \theta =\frac{x}{r}[/latex] and [latex]\sin \theta =\frac{y}{r}[/latex].

[latex]\begin{align}z&=13\left(\cos \theta +i\sin \theta \right) \\ &=13\left(\frac{12}{13}+\frac{5}{13}i\right) \\ &=12+5i \end{align}[/latex]

The rectangular form of the given number in complex form is [latex]12+5i[/latex].