z_1 = (-1 - i 5) z1=(−1−i5)
r_1 = sqrt(1^2 + 5^2) = sqrt26r1=√12+52=√26
theta_1 = arctan (-5/-1) = arctan 5 = 258.69^@, " III Quadrant"θ1=arctan(−5−1)=arctan5=258.69∘, III Quadrant
z_2 = (3 - i 6)z2=(3−i6)
r_2 = sqrt(3^2 + 6^2) = sqrt45r2=√32+62=√45
theta_2 = -6/3 = - 2 = 296.57^@, " IV Quadrant"θ2=−63=−2=296.57∘, IV Quadrant
z_1 * z_2 = (r_1 * r_2) * (cos (theta-1 + theta_2) + i sin (theta_1 + theta_2))z1⋅z2=(r1⋅r2)⋅(cos(θ−1+θ2)+isin(θ1+θ2))
z_1 * z_2 = (sqrt 26 * sqrt 45) * (cos (258.69 + 296.57) + i sin(258.69 + 296.57))z1⋅z2=(√26⋅√45)⋅(cos(258.69+296.57)+isin(258.69+296.57))
=> 34.21 (cos 555.26 + i sin 555.26) = 34.21(-0.9647 - i 0.2632)⇒34.21(cos555.26+isin555.26)=34.21(−0.9647−i0.2632)
color(crimson)((-1-i5) (3 - i 6) = (-33 - i 9)(−1−i5)(3−i6)=(−33−i9)
