Exercícios Números Complexos

Regarding task 1:

r = √(4² + 5²)  =  6,4      ¦      φ = arctan(-5/-4) = 51,34°

Since the real and imaginary parts are both negative, Z lies in the 3rd quadrant => φ = 180° + arctan(-5/-4) = 231,34 °

Exercises Complex numbers

Regarding task 2:

First way to solution:
Z = Z₁ Z₂ = (4 – i)(3 + 4i) = 12 +16i -3i - 4i² = 16 + 13i

Second way to solution:
r₁ =  √(4² + 1²)  = √17   => r = r₁ * r₂ = 5√17  = 20,62
r₂ = √(3² + 4²)  = 5

φ₁ = arctan(-1/4) = -14,04°  => φ = φ₁ + φ₂ = -14,04° + 53,09° = 39,09°
φ₂ = arctan(4/3) = 53,09°

=>   Z = 20,62 e^39,09°

Note: Pointer Z1 lies in the 4th quadrant, because real part > 0 and imaginary part < 0, therefore φ1 remains negative and no 180° must be added to it.

Test:
Z = 20,62 [cos39,09° + sin39,09°i] = 16 + 13i