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

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

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

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

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

thanks

To find the modulus of a complex number, we can use the formula:

|z| = √(Re(z)^2 + Im(z)^2)

where Re(z) is the real part of the complex number and Im(z) is the imaginary part of the complex number.

Let's find the modulus of the complex number 15-4i:

Re(z) = 15
Im(z) = -4

|z| = √(15^2 + (-4)^2)
= √(225 + 16)
= √241

Therefore, the modulus of 15-4i is √241.

To find the argument of a complex number, we can use the formula:

arg(z) = arctan(Im(z)/Re(z))

Let's find the argument of the complex number 15-4i:

Re(z) = 15
Im(z) = -4

arg(z) = arctan((-4)/15)
= arctan(-4/15)

The argument is the angle in radians from the positive real axis to the complex number in the complex plane. To get the value in degrees, we can convert the angle from radians to degrees:

arg(z) in degrees = arg(z) in radians * (180/π)

Therefore, the argument of 15-4i in degrees is arctan(-4/15) * (180/π).

For the complex number a-ai where a > 0, the modulus and argument can be found in a similar manner. Let's say a = b, where b is a positive real number greater than 0.

Re(z) = b
Im(z) = -b

|z| = √(b^2 + (-b)^2)
= √(b^2 + b^2)
= √(2b^2)
= b√2

arg(z) = arctan((-b)/b)
= arctan(-1)
= -π/4

Therefore, the modulus of a-ai is b√2, and the argument is -45 degrees (-π/4 radians).