## Precalculus (6th Edition) Blitzer

\begin{align} & \frac{\left[ 2\left( \cos \,60{}^\circ +i\,\sin \,60{}^\circ \right) \right]\left[ \sqrt{2}\left( \cos \,315{}^\circ +i\,\sin \,315{}^\circ \right) \right]}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]}; \\ & \frac{\sqrt{2}}{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right);\ \approx \frac{1}{2}+\frac{i}{2} \\ \end{align}
Consider the provided expression $z=\frac{\left( 1+i\sqrt{3} \right)\left( 1-i \right)}{\left( 2\sqrt{3}-2i \right)}$ First convert it into polar form: \begin{align} & z=\frac{\left( 1+i\sqrt{3} \right)\left( 1-i \right)}{\left( 2\sqrt{3}-2i \right)} \\ & =\frac{\left[ 2\left( \cos \,60{}^\circ +i\,\sin \,60{}^\circ \right) \right]\left[ \sqrt{2}\left( \cos \,315{}^\circ +i\,\sin \,315{}^\circ \right) \right]}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]} \end{align} Apply the multiplication rule of complex numbers in the numerator of the above expression: $z=\frac{\left( \sqrt{2}\times 2 \right)\left( \cos \left( 60{}^\circ +315{}^\circ \right)+i\,\sin \left( 60{}^\circ +315{}^\circ \right) \right)}{\left[ 2\times 2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right) \right]}$ Simplify the above expression: $z=\frac{\sqrt{2}\left( \cos \,375{}^\circ +i\,\sin \,375{}^\circ \right)}{2\left( \cos \,330{}^\circ +i\,\sin \,330{}^\circ \right)}$ Apply the quotient rule of complex numbers in the above expression: \begin{align} & z=\frac{\sqrt{2}\left( \cos \left( 375{}^\circ -330{}^\circ \right)+i\,\sin \left( 375{}^\circ -330{}^\circ \right) \right)}{2} \\ & =\frac{\sqrt{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right)}{2} \end{align} The above expression is the polar form of the provided expression. Convert it into rectangular form by substituting the values of $\cos \,45{}^\circ$ and $\sin \,45{}^\circ$ in the above expression: \begin{align} & z=\frac{\sqrt{2}}{2}\left( \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right) \\ & =\frac{\sqrt{2}}{2\sqrt{2}}\left( 1+i \right) \end{align} Simplify the above equation: $z=\left( \frac{1}{2}+\frac{i}{2} \right)$ The above expression is the rectangular form of the provided expression. The polar form of the provided expression is $\frac{\sqrt{2}}{2}\left( \cos \,45{}^\circ +i\,\sin \,45{}^\circ \right)$. The rectangular form of the provided expression is $\approx \frac{1}{2}+\frac{i}{2}$.