On the complex plane, the number [latex]z=4i[/latex] is the same as [latex]z=0+4i[/latex]. Writing it in polar form, we have to calculate [latex]r[/latex] first.

[latex]\begin{align}&r=\sqrt{{x}^{2}+{y}^{2}} \\ &r=\sqrt{{0}^{2}+{4}^{2}} \\ &r=\sqrt{16} \\ &r=4 \end{align}[/latex]

Next, we look at [latex]x[/latex]. If [latex]x=r\cos \theta[/latex], and [latex]x=0[/latex], then [latex]\theta =\frac{\pi }{2}[/latex]. In polar coordinates, the complex number [latex]z=0+4i[/latex] can be written as [latex]z=4\left(\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)\right)[/latex] or [latex]4\text{cis}\left(\frac{\pi }{2}\right)[/latex].

First, find the value of [latex]r[/latex].

[latex]\begin{align}&r=\sqrt{{x}^{2}+{y}^{2}} \\ &r=\sqrt{{\left(-4\right)}^{2}+\left({4}^{2}\right)} \\ &r=\sqrt{32} \\ &r=4\sqrt{2} \end{align}[/latex]

Find the angle [latex]\theta[/latex] using the formula:

[latex]\begin{align}&\cos \theta =\frac{x}{r} \\ &\cos \theta =\frac{-4}{4\sqrt{2}} \\ &\cos \theta =-\frac{1}{\sqrt{2}} \\ &\theta ={\cos }^{-1}\left(-\frac{1}{\sqrt{2}}\right)=\frac{3\pi }{4} \end{align}[/latex]

Thus, the solution is [latex]4\sqrt{2}\cos\left(\frac{3\pi }{4}\right)[/latex].