1. **Problem Statement:** Given two complex numbers in polar form, $z_1 = r_1(\cos \theta + i \sin \theta)$ and $z_2 = r_2(\cos \varphi + i \sin \varphi)$, we want to understand how to multiply and divide them using their moduli and arguments. 2. **Polar Form of a Complex Number:** A complex number $z = a + bi$ can be written as $$z = r(\cos \theta + i \sin \theta)$$ where $r = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \tan^{-1}(\frac{b}{a})$ is the argument. 3. **Multiplication Rule:** When multiplying two complex numbers in polar form: - Multiply their moduli: $r = r_1 \times r_2$ - Add their arguments: $\theta = \theta_1 + \theta_2$ 4. **Derivation of Multiplication:** Starting with $$z_1 z_2 = r_1(\cos \theta_1 + i \sin \theta_1) \times r_2(\cos \theta_2 + i \sin \theta_2)$$ Expanding, $$= r_1 r_2 [ (\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2) + i (\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2) ]$$ Using trig identities, $$= r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))$$ 5. **Example of Multiplication:** Given $z_1 = 3(\cos 30^\circ + i \sin 30^\circ)$ and $z_2 = 4(\cos 60^\circ + i \sin 60^\circ)$, $$r = 3 \times 4 = 12$$ $$\theta = 30^\circ + 60^\circ = 90^\circ$$ So, $$z_1 z_2 = 12(\cos 90^\circ + i \sin 90^\circ)$$ 6. **Division Rule:** When dividing two complex numbers in polar form: - Divide their moduli: $r = \frac{r_1}{r_2}$ - Subtract the argument of the denominator from the numerator: $\theta = \theta_1 - \theta_2$ 7. **Derivation of Division:** $$\frac{z_1}{z_2} = \frac{r_1(\cos \theta_1 + i \sin \theta_1)}{r_2(\cos \theta_2 + i \sin \theta_2)} = \frac{r_1}{r_2} \times \frac{\cos \theta_1 + i \sin \theta_1}{\cos \theta_2 + i \sin \theta_2}$$ Multiply numerator and denominator by the conjugate of denominator: $$= \frac{r_1}{r_2} \times \frac{(\cos \theta_1 + i \sin \theta_1)(\cos \theta_2 - i \sin \theta_2)}{(\cos \theta_2 + i \sin \theta_2)(\cos \theta_2 - i \sin \theta_2)}$$ Simplify denominator: $$= \frac{r_1}{r_2} \times \frac{\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)}{1}$$ 8. **Final Polar Form for Division:** $$\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))$$ **Summary:** - Multiplying complex numbers in polar form multiplies moduli and adds arguments. - Dividing complex numbers in polar form divides moduli and subtracts arguments. This method simplifies complex number operations by converting multiplication and division into simple arithmetic on moduli and arguments.