Respuesta :
Answer:
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!!!