# Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 799: 63

The rectangular form of $z=6\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)$ is $z=\left( -3+3i\sqrt{3} \right)$.

#### Work Step by Step

The rectangular form of a complex number is $z=a+ib$, where $a$ and $b$ are rectangular coordinates. In the polar form the complex number $z=a+ib$ is represented as $z=r\left( \cos \theta +i\sin \theta \right)$, where r is the distance of the complex number from the origin and $\theta$ is the respective angle. Here, the given complex number is in the polar form with $r=6$ and $\theta =\frac{2\pi }{3}$. Since, $\cos \frac{2\pi }{3}=-\frac{1}{2}\ \text{ and }\ \sin \frac{2\pi }{3}=\frac{\sqrt{3}}{2}\$ Therefore, \begin{align} & z=6\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right) \\ & =6\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right) \\ & =\left( -\frac{6}{2}+i\frac{6\sqrt{3}}{2} \right) \\ & =\left( -3+3i\sqrt{3} \right) \end{align}

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