Precalculus (6th Edition) Blitzer

The quotient of two complex numbers written in polar form can be found by the use of the quotient rule of complex numbers. In this method, the modulus and argument of the provided complex numbers are divided and subtracted respectively. For example, the quotient of two complex numbers ${{z}_{1}}={{r}_{1}}\left( \cos \theta +i\sin \theta \right)$ and ${{z}_{2}}={{r}_{2}}\left( \cos \alpha +i\sin \alpha \right)$ can be found by the use of the quotient rule as shown below: $\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{{{r}_{1}}}{{{r}_{2}}}\left[ \cos \left( \theta -\alpha \right)+i\sin \left( \theta -\alpha \right) \right]$ Above is the quotient of two complex numbers provided in polar form by the use of the quotient rule. Example: The above statement can be justified with the help of an example. Let ${{z}_{1}}=5\left( \cos 30{}^\circ +i\sin 30{}^\circ \right)$ and ${{z}_{1}}=10\left( \cos 60{}^\circ +i\sin 60{}^\circ \right)$ be two complex numbers provided in polar form. Then by the use of quotient rule of complex numbers, the provided two numbers can be divided as shown below: \begin{align} & \frac{{{z}_{1}}}{{{z}_{2}}}=\frac{5}{10}\left[ \cos \left( 30{}^\circ -60{}^\circ \right)+i\sin \left( 30{}^\circ -60{}^\circ \right) \right] \\ & =\frac{1}{2}\left[ \cos \left( -30{}^\circ \right)+i\sin \left( -30{}^\circ \right) \right] \end{align} Applying the property of cosine and sine functions $\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{1}{2}\left[ \cos 30{}^\circ -i\sin 30{}^\circ \right]$ Above is the quotient of two complex numbers provided in polar form by the use of quotient rule.