# Problems from Complex numbers in trigonometric form: product and quotient

Calculate: $$\dfrac{21\cdot[\cos(225^\circ)+i\cdot \sin(225^\circ)]}{9\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]}$$

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### Development:

$$\dfrac{21\cdot[\cos(225^\circ)+i\cdot \sin(225^\circ)]}{9\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]}=\dfrac{27}{9}\cdot [\cos(225^\circ-180^\circ)+i\cdot\sin(225^\circ-180^\circ)]$$

$$=3\cdot [\cos(45^\circ)+i\cdot\sin(45^\circ)]=3\cdot e^{i45^\circ}$$

### Solution:

$$3\cdot e^{i45^\circ}$$

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Write the following complex numbers in the trigonometric form:

• $$3+3i$$
• $$5_{180^\circ}$$
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### Development:

• 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: $$z=5\cdot[\cos(180^\circ)+i\cdot \sin(180^\circ)]=5\cdot e^{i180^\circ}$$\$

### Solution:

• $$z=\sqrt{18}\cdot e^{i45^\circ}$$
• $$z=5\cdot e^{i180^\circ}$$
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