Complex numbers in trigonometric form: product and quotient
Write the following complex numbers in the trigonometric form:
- $3+3i$
- $5_{180^\circ}$
First we change it to polar form $$\displaystyle z=3+3i \ \Rightarrow \ \left\{ \begin{array}{l} |z|=\sqrt{3^2+3^2}=\sqrt{18} \\ \alpha=\arctan\big( \dfrac{3}{3} \big) =45^\circ \end{array} \right\} \Rightarrow \ z=\sqrt{18}_{45^\circ}$$
And now we calculate the trigonometric form: $$z=\sqrt{18}\cdot[\cos(45^\circ)+i\cdot \sin(45^\circ)]=\sqrt{18}\cdot e^{i45^\circ}$$
Since it is already written in polar form it is straight forward that: $$z=5\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]=5\cdot e^{i180^\circ}$$
- $z=\sqrt{18}\cdot e^{i45^\circ}$
- $z=5\cdot e^{i180^\circ}$
Calculate: $\dfrac{21\cdot[\cos(225^\circ)+i\cdot \sin(225^\circ)]}{9\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]}$
$ \dfrac{21\cdot[\cos(225^\circ)+i\cdot \sin(225^\circ)]}{9\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]}=\dfrac{27}{9}\cdot [\cos(225^\circ-180^\circ)+i\cdot\sin(225^\circ-180^\circ)]$
$ =3\cdot [\cos(45^\circ)+i\cdot\sin(45^\circ)]=3\cdot e^{i45^\circ}$
$$3\cdot e^{i45^\circ}$$