Question

1. Find the modulus and argument of the following complex numbers, and write them in trigonometric form:

a. 5 – 8i

Answer = Sqrt{89} ( cos (-1.01219701) + i sin ( -1.01219701))

b. –1 – i

answer = Sqrt2 (cos (5pi/4) + i sin (5pi/4))

c. –5 + 12i

answer = 13 (cos ( -4.31759786068493) + i sin ( -4.31759786068493))

d. 1 + sqrt{3}i

answer = 2 ( cos (pi/3) + i sin (pi/3))

Answer 1

all look good, but I'm curious why you had negative angles for (a) annd (c).
For (a) I could see it, because the angle is greater than -pi
For (c) I would have added 2pi to get an angle less than pi, since the angle is in QII.

Answer 2

I think he/she has a negative angle for a) by entering the data into a calculator and simply blindly accepting the answer the calculator gives.
Remember the calculator is programmed to give the closest angle to zero.
so tan^-1 (-8/5) = -1.012....

I would take 2π - 1.012....