A complex number z = a + bi can be written using r = |z| and θ = arg z (argument) as
z = r(cosθ + i sinθ). This form is called the polar form.
Example ① Express z = 1 + i in polar form.
Solution:r = √2、tan θ = 1 → θ = π/4.
z = √2(cos(π/4) + i sin(π/4))
Solution:r = √2、tan θ = 1 → θ = π/4.
z = √2(cos(π/4) + i sin(π/4))
Key Point
- Adjust the argument according to the quadrant.
Practice Problems
- Express z = −√3 + i in polar form.
Solution and Explanation
- Answer:2(cos(5π/6) + i sin(5π/6))