Find zw and zw Leave your answers in polar form z 9cos 110

Find zw and z/w. Leave your answers in polar form. z = 9(cos 110 degree + i sin 110 degree) w = 2 cos 210 degree + i sin 210 degree) What is the product? ___ [cos ___ degree + i sin ___ degree] (Simplify your answers. Type any angle measures in degrees. Use angle measures greater than or equal to 0 and less than 360.) What is the quotient? __ [cos __ degree + i sin ___ degree] (Simplify your answers. Type any angle measures in degrees. Use angle measures greater than or equal to 0 and less than 360.)

Solution

We will use euler\'s theorem

z = r (cos + i sin ) = r eⁱ

z = 9 (cos 110⁰ + i sin 110⁰) = 9 e^{i 110}

w = 2 (cos 210⁰ + i sin 210⁰) = 2 e^{i 210}

zw = 9 e^{i 110}  2 e^{i 210}

zw = 18 e^{i (110+210)}

zw = 18 e^{i (320)}

zw = 18 (cos 320⁰ + i sin 320⁰)

z/w = (9 e^{i 110}) / (2 e^{i 210})

z/w = 4.5 e^{i (110 - 210)}

z/w = 4.5 e^{i (-100)}

z/w = 4.5 (cos -100⁰ + i sin -100⁰)

-100⁰ = -100⁰ + 360⁰= 260⁰

z/w = 4.5 (cos 260⁰ + i sin 260⁰)