Express the complex number in trigonometric form.

Question

Express the complex number in trigonometric form. 3 - 3i. 

Answer 1

z = 3 - 3i

 

z = |z| e^iφ

 

e^iφ = cosφ + i sinφ

 

φ = tan^-1[Im(z) / Re(z)]

 

Im(z) = -3

 

Re(z) = 3

 

φ = tan^-1(-3 / 3) = 7π/4

 

|z| = √[Re(z)² + Im(z)²] = √[(3)² + (-3)²] = √(2*9) = 3√2

 

z = 3√2 e^i7π/4

 

z = 3√2 [cos(7π/4) + i sin (7π/4)]