Find zw and StartFraction z Over w EndFraction . Leave your answers in polar form. z equals 4 (cosine 150 degrees plus i sine 150 degrees )w equals 2 (cosine 250 degrees plus i sine 250 degrees )

a) zw = 8 (Cos 40° + i Sin 40°)

b) (z/w) = 2 (Cos 260° + i Sin 260°)

Step-by-step explanation:

z = 4(Cos 150° + i Sin 150°)

w = 2 (Cos 250° + i Sin 250°)

To first simplify,

Cos 150° = -0.8660

Sin 150° = 0.50

Cos 250° = -0.3420

Sin 250° = -0.9397

z = 4(Cos 150° + i Sin 150°)

z = 4 (-0.866 + 0.5i)

z = (-3.464 + 2i)

w = 2 (Cos 250° + i Sin 250°)

w = 2 (-0.342 -0.9397i)

w = (-0.684 - 1.8794i)

a) zw = (-3.464 + 2i) (-0.684 - 1.8794i)

zw = 2.369376 + 6.5102416i - 1.368i - 3.7588i²

Note that i² = -1

zw = 2.369376 + 5.1422416i + 3.7588

zw = (6.128176 + 5.1422416i)

A general complex number z = x + it has the Polar form = r (cos θ + i sin θ)

r = √(x² + y²)

θ = arctan (y/x)

zw = (6.128176 + 5.1422416i)

x = 6.128176

y = 5.1422416

r = √(6.128176² + 5.1422416²) = 7.99997 = 8

θ = arctan (5.1422416/6.128176) = 40°

zw = 8 (Cos 40° + i Sin 40°)

b) (z/w) = (-3.464 + 2i) / (-0.684 - 1.8794i)

To simplify This, we first rationalize, that is, multiply numerator and denominator by (-0.684 + 1.8794i)

(z/w) = [(-3.464 + 2i)×(-0.684 + 1.8794i)] ÷ [((-0.684 - 1.8794i)×((-0.684 + 1.8794i)

(z/w) = [2.369376 - 6.5102416i - 1.368i + 3.7588i²] ÷ [0.467856 - 3.53214436i²]

Note that i² = -1

(z/w) = [2.369376 - 3.7588 - 7.8782416i] ÷ [0.467856 + 3.53214436]

(z/w) = (-1.389424 - 7.8782416i)/4

(z/w) = (-0.347356 - 1.9695604i)

x = -0.347356

y = -1.9695604

r = √[(-0.347356)² + (-1.9695604)²] = 1.9997 = 2

θ = arctan (-1.9695604)/(-0.347356) = 80° on the first quadrant, but the signs on x and y indicates that this is the third quadrant, hence

θ = 180° + 80° = 260°

(z/w) = 2 (Cos 260° + i Sin 260°)

Hope this Helps!!!

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