## Precalculus (6th Edition) Blitzer

The rectangular form of the complex number is $-2\sqrt{3}+2i$.
Consider any complex number, given by $z=x+iy$; for a complex number in rectangular form, $z=x+iy$ …… (1) The polar form is given by, $z=r\left( \cos \theta +i\sin \theta \right)$ …… (2) Here, $x=r\cos \theta \ \text{ and }\ y=r\sin \theta$, dividing the value of y by x, to get, $\tan \theta =\frac{y}{x}$ Also, the value of r is called as moduli of the complex number, given by, $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ For any complex number in polar form, $z=r\left( \cos \theta +i\sin \theta \right)$, the rectangular form is, Using (1) and (2), \begin{align} & z=4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right) \\ & z=x+iy \\ \end{align} Simplify it further to get, \begin{align} & 4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right)=4\cos \frac{5\pi }{6}+4i\sin \frac{5\pi }{6} \\ & =\left( 4\times -\frac{\sqrt{3}}{2} \right)+i\left( 8\times -\frac{1}{\sqrt{2}} \right) \\ & =-2\sqrt{3}+2i \\ & 4\left( \cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6} \right)=-2\sqrt{3}+2i \end{align} The rectangular form of the complex number is $-2\sqrt{3}+2i$. Hence, the rectangular form of the complex number is $-2\sqrt{3}+2i$.