Find zw and z/w?

Question

Find zw and z/w. Leave your answers in polar form.

(1) z = 4[cos(3pi/8) + isin(3pi/8)]
w = 2[cos(9pi/16) + isin(9pi/16)]

(2) z = 1 - i
w= 1 - sqrt{3}i

Answer 1

Remember that when multiplying complex numbers in polar form that the moduli multiply and the arguments add. So r(cos θ+ i sin θ) * s(cos φ + i sin φ) = rs(cos (θ+φ) + i sin (θ+φ)). Thus in particular, we have:

#1a: zw = 8(cos (15π/16) + i sin (15π/16))

Similarly, when dividing complex numbers, it's just the opposite of multiplying them -- moduli divide and arguments subtract. So in particular:

#1b: zw = 2(cos (-3π/16) + i sin (-3π/16))

#2: I tend to prefer to leave stuff that starts in rectangular form in rectangular form, but we can do it both ways:

zw = (1-i)(1-i√3) = (1-√3) - i (1+√3)
z/w = (1-i)/(1-i√3) = (1-i)(1+i√3) / (1+3) = (1+√3 + i(√3-1)) / 4 = (1/4 + √3/4) + i (√3/4 - 1/4)

Or in polar form:

|z| = √2, arg (z) = -π/4
|w| = 2, arg (w) = -π/3

zw = 2√2 (cos (-7π/12) + i sin (-7π/12))
z/w = 1/√2 (cos (π/12) + i sin (π/12))