Precalculus (6th Edition) Blitzer

Published by Pearson

Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 768: 77

Answer

\begin{align} & \left[ 1\left( \cos \,90{}^\circ +i\,\sin \,90{}^\circ \right) \right]\left[ 2\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right) \right]\left[ 2\left( \cos \,150{}^\circ +i\,\sin \,150{}^\circ \right) \right]; \\ & 4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right);\ \approx 1.4641-5.4641i \\ \end{align}

Work Step by Step

Consider the provided expression $z=i\left( 2+2i \right)\left( -\sqrt{3}+i \right)$ First convert it into polar form: \begin{align} & z=i\left( 2+2i \right)\left( -\sqrt{3}+i \right) \\ & =\left[ 1\left( \cos \,90{}^\circ +i\,\sin \,90{}^\circ \right) \right]\left[ 2\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right) \right]\left[ 2\left( \cos \,150{}^\circ +i\,\sin \,150{}^\circ \right) \right] \end{align} Apply the multiplication rule of complex numbers: \begin{align} & z=\left( 1\times 2\sqrt{2}\times 2 \right)\left( \cos \left( 90{}^\circ +45{}^\circ +150{}^\circ \right)+i\,\sin \left( 90{}^\circ +45{}^\circ +150{}^\circ \right) \right) \\ & =4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right) \end{align} The above expression is the polar form of the provided expression. Convert it into rectangular form: $z=4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right)$ Substitute the values of $\cos \,285{}^\circ$ and $\sin \,285{}^\circ$ in the above expression: \begin{align} & z=4\sqrt{2}\left( 0.2588-i0.9659 \right) \\ & =1.4641-i5.4641 \end{align} The above expression is the rectangular form of the provided expression is: The polar form of the provided expression is $4\sqrt{2}\left( \cos \,285{}^\circ +i\,\sin \,285{}^\circ \right)$. The rectangular form of the provided expression is $\approx 1.4641-i5.4641$.

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