1. **Problem Statement:** Understand the polar form of a complex number and how to multiply and divide complex numbers using this form. 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 (distance from origin). - $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ is the argument (angle with positive real axis). 3. **Multiplying Complex Numbers in Polar Form:** - Multiply the moduli: $r_1 r_2$ - Add the arguments: $\theta_1 + \theta_2$ So, if $$z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$$ and $$z_2 = r_2(\cos \theta_2 + i \sin \theta_2),$$ then $$z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)\right).$$ 4. **Example of Multiplication:** Given $$z_1 = 3(\cos 30^\circ + i \sin 30^\circ)$$ and $$z_2 = 4(\cos 60^\circ + i \sin 60^\circ),$$ - Multiply moduli: $3 \times 4 = 12$ - Add arguments: $30^\circ + 60^\circ = 90^\circ$ So, $$z_1 z_2 = 12(\cos 90^\circ + i \sin 90^\circ) = 12(0 + i \times 1) = 12i.$$ 5. **Dividing Complex Numbers in Polar Form:** - Divide the moduli: $\frac{r_1}{r_2}$ - Subtract the arguments: $\theta_1 - \theta_2$ If $$z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$$ and $$z_2 = r_2(\cos \theta_2 + i \sin \theta_2),$$ then $$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right).$$ 6. **Example of Division:** Given $$z_1 = 5(\cos 120^\circ + i \sin 120^\circ)$$ and $$z_2 = 2(\cos 45^\circ + i \sin 45^\circ),$$ - Divide moduli: $\frac{5}{2} = 2.5$ - Subtract arguments: $120^\circ - 45^\circ = 75^\circ$ So, $$\frac{z_1}{z_2} = 2.5(\cos 75^\circ + i \sin 75^\circ).$$ This form makes multiplication and division of complex numbers easier by working with magnitudes and angles instead of real and imaginary parts.