1. Planteamos el problema: Calcular \( zw \) para los casos dados, usando las fórmulas de multiplicación de números complejos en forma polar: \( z = r_1(\cos \theta_1 + i \sin \theta_1) \), \( w = r_2(\cos \theta_2 + i \sin \theta_2) \). El producto es \( zw = r_1 r_2 \big( \cos(\theta_1+\theta_2) + i \sin(\theta_1+\theta_2) \big) \). 2. Para el cociente de números complejos en forma polar: \( \frac{z}{w} = \frac{r_1}{r_2} \big( \cos(\theta_1-\theta_2) + i \sin(\theta_1-\theta_2) \big) \). --- **Multiplicación:** 39. \( z = \cos 17^\circ + i \sin 17^\circ, w = 8(\cos 13^\circ + i \sin 13^\circ) \) \[ r_z = 1, \theta_z = 17^\circ; \quad r_w=8, \theta_w=13^\circ \] \[ zw = 1 \times 8 \big( \cos(17^\circ + 13^\circ) + i \sin(17^\circ + 13^\circ) \big) = 8 (\cos 30^\circ + i \sin 30^\circ) \] 40. \( z = 4 (\cos 25^\circ + i \sin 25^\circ), w = 7(\cos 35^\circ + i \sin 35^\circ) \) \[ zw = 4 \times 7 \big( \cos(25^\circ + 35^\circ) + i \sin(25^\circ + 35^\circ) \big) = 28 (\cos 60^\circ + i \sin 60^\circ) \] 41. \( z = 6 (\cos 100^\circ + i \sin 100^\circ), w = 2(\cos 50^\circ + i \sin 50^\circ) \) \[ zw = 6 \times 2 \big( \cos(100^\circ + 50^\circ) + i \sin(100^\circ + 50^\circ) \big) = 12 (\cos 150^\circ + i \sin 150^\circ) \] 42. \( z = 9 (\cos 110^\circ + i \sin 110^\circ), w = \cos 160^\circ + i \sin 160^\circ \) (asumiendo \( r_w=1 \)) \[ zw = 9 \times 1 \big( \cos(110^\circ + 160^\circ) + i \sin(110^\circ + 160^\circ) \big) = 9 (\cos 270^\circ + i \sin 270^\circ) \] --- **División:** 43. \( z = \cos 208^\circ + i \sin 208^\circ, w = \cos 73^\circ + i \sin 73^\circ \) \[ r_z=1, r_w=1 \] \[ \frac{z}{w} = \frac{1}{1} \big( \cos(208^\circ - 73^\circ) + i \sin(208^\circ - 73^\circ) \big) = \cos 135^\circ + i \sin 135^\circ \] 44. \( z = 6 (\cos 70^\circ + i \sin 70^\circ), w = 3 (\cos 40^\circ + i \sin 40^\circ) \) \[ \frac{z}{w} = \frac{6}{3} \big( \cos(70^\circ - 40^\circ) + i \sin(70^\circ - 40^\circ) \big) = 2 (\cos 30^\circ + i \sin 30^\circ) \] 45. \( z = \cos 100^\circ + i \sin 100^\circ, w = 7 (\cos 10^\circ + i \sin 10^\circ) \) \[ \frac{z}{w} = \frac{1}{7} \big( \cos(100^\circ - 10^\circ) + i \sin(100^\circ - 10^\circ) \big) = \frac{1}{7} (\cos 90^\circ + i \sin 90^\circ) \] 46. \( z = 9 (\cos 200^\circ + i \sin 200^\circ), w = 3 (\cos 20^\circ + i \sin 20^\circ) \) \[ \frac{z}{w} = \frac{9}{3} \big( \cos(200^\circ - 20^\circ) + i \sin(200^\circ - 20^\circ) \big) = 3 (\cos 180^\circ + i \sin 180^\circ) \] --- Final: Multiplicación: \(39. \quad 8(\cos 30^\circ + i \sin 30^\circ)\) \(40. \quad 28(\cos 60^\circ + i \sin 60^\circ)\) \(41. \quad 12(\cos 150^\circ + i \sin 150^\circ)\) \(42. \quad 9(\cos 270^\circ + i \sin 270^\circ)\) División: \(43. \quad \cos 135^\circ + i \sin 135^\circ\) \(44. \quad 2(\cos 30^\circ + i \sin 30^\circ)\) \(45. \quad \frac{1}{7}(\cos 90^\circ + i \sin 90^\circ)\) \(46. \quad 3(\cos 180^\circ + i \sin 180^\circ)\)