How do you divide #( 3i-2) / (i +4 )# in trigonometric form?

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Answer 1 · 2018-07-08T01:30:36

#color(maroon)((-2 + 3 i) / (4 + i) ~~ -0.2931 - i 0.8231#

To divide #(-2 + 3 i) / (4 + i)# using trigonometric form.

#z_1 = (-2 + 3 i), z_2 = (4 + i)#

#r_1 = sqrt(-2^2 + 3^2) = sqrt 13

#r_2 = sqrt(4^2 + 1^2) = sqrt 17#

#theta_1 = arctan (-2/3) = 146.31^@, " II quadrant"#

#Theta_2 = arctan(4/1) = 75.96^@, " I quadrant"#

#z_1 / z_2 = (r_1 / r_2) * (cos (theta_1 - theta_2) + i sin (theta_1 - theta_2))#

#z_1 / z_2 = sqrt(13/17) * (cos (146.31 - 75.96 ) + i sin (146.31 - 75.96 ))#

#z_1 / z_2 = sqrt(13/17) * (cos (70.41) + i sin (70.41))#

#color(maroon)((-2 + 3 i) / (4 + i) ~~ -0.2931 - i 0.8231#