# Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 767: 35

The rectangular form of the given complex number is $-18.2-8.5i$.

#### Work Step by Step

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, $x=r\cos \theta \ \text{ and }\ y=r\sin \theta$, Divide 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=20\left( \cos 205{}^\circ +i\sin 205{}^\circ \right) \\ & z=x+iy \\ \end{align} Simplify it further to get, \begin{align} & 20\left( \cos 205{}^\circ +i\sin 205{}^\circ \right)=20\cos 205{}^\circ +20i\sin 205{}^\circ \\ & =\left( 20\times -0.91 \right)+i\left( 20\times -0.42 \right) \\ & =-18.2-8.5i \\ & 20\left( \cos 205{}^\circ +i\sin 205{}^\circ \right)=-18.2-8.5i \end{align} The rectangular form of the complex number is $-18.2-8.5i$.

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