## Precalculus (6th Edition) Blitzer

The rectangular form of the given complex number is $-7i$.
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=7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right) \\ & z=x+iy \\ \end{align} Simplify it further to get, \begin{align} & 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=7\cos \frac{3\pi }{2}+7i\sin \frac{3\pi }{2} \\ & =\left( 7\times 0 \right)+i\left( 7\times -1 \right) \\ & =-7i \\ & 7\left( \cos \frac{3\pi }{2}+i\sin \frac{3\pi }{2} \right)=-7i \end{align} The rectangular form of the complex number is $-7i$