Question

How do you divide `( 7i+5) / ( -3i +8 )` in trigonometric form?

Answer 1

`color(blue)((5 + 7i) / (8 - 3i) ~~ 0.2603 - i 0.9728`

Explanation:

To divide `(5 +7 i) / (8 - 3i)` using trigonometric form.

`z_1 = (5 +7 i), z_2 = (8 - 3i)`

#r_1 = sqrt(5^2 + 7^2) = sqrt 74

`r_2 = sqrt(8^2 + -3^2) = sqrt 73`

`theta_1 = arctan (7/5) = 54.46^@, " I quadrant"`

`Theta_2 = arctan(-3/8) = 339.44^@, " IV 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(74/73) * (cos (54.46 - 339.44 ) + i sin (54.46 - 339.44 ))`

`z_1 / z_2 = 1.007 * (cos (-284.98) + i sin (-284.98)) = 1.007 (0.2585 - i 0.966)`

`color(blue)((5 + 7i) / (8 - 3i) ~~ 0.2603 - i 0.9728`