find the modulus and the argument of the following complex numbers:

Question

15-4i and a-ai where a is greater than 0

Answer 1

if z = a + b i
r^2 = sqrt( a^2 + b^2)
and a = r cos T
and b = r sin T

so
if a = 15
and b = -4
then r^2 = 15^2 + (-4)^2 = 225+16
r = 15.524
cos T = 15/15.524
so T = 14.9 or -14.9
sin T = -4/15.52
so T = -14.9 or (180-14.9)=165.1
T = -14.9 which is +345.1 in the fourth quadrant satisfies both, so that is it. (Besides we could see that 15-4i was in quadrant 4)

r = sqrt (2 a^2) = a sqrt 2
fourth quadrant again
so
sin T = -1/sqrt (2) = = -.707
T = -45 degrees which is + 315 degrees

Answer 2

im sorry they were two different questions you have to find the modulus and the argument
ex.15-4i
modulus-4
but i don't know what the argument is

Answer 3

I did them as two different questions
for #1
modulus = r = 15.524
argument = T (for theta) = -14.9 degrees (or +345.1 degrees)

for #2
modulus = a sqrt 2
theta = argument = -45 (or 315) degrees

Answer 4

thanks