Find zw and z/w?

Question

For each question below, find zw and z/w. Leave your answer in polar form.

NOTE: d = degrees. Okay?

(1) z = cos120d + isin120d; w = cos100d + isin100d

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

Answer 1

If you have two complex numbers in polar form:

a * (cosb + isinb) = (a * e^(i*b)) = (a,b)
c * (cosd + isind) = (c * e^(i*d)) = (c,d)

... then the product also in polar form is (a*c, b+d), and the quotient is (a/c, b-d)

You multiply or divide the magnitude, and add or subtract the direction.

(1) z = cos120d + isin120d; w = cos100d + isin100d

z = cos120d + isin120d
z = 1 * (cos120 + isin120)
z = 1 * (cos(2pi/3) + isin(2pi/3))
z = (1, 2pi/3)

Similarly, w = (1, 5pi/9)

product = (1*1, 2pi/3 + 5pi/9) = (1, 11pi/9)
product = (1, 220) converted back to degrees

quotient = (1/1, 2pi/3 - 5pi/9) = (1, pi/9)
quotient = (1, 20) converted back to degrees

(2) z = (4, 3pi/8), w = (2, 9pi/16)

product = (4*2, 3pi/8 + 9pi/16) = (8, 15pi/16)
quotient = (4/2, 3pi/8 - 9pi/16) = (2, -3pi/16)