Lesson 3 argument polar form of a complex number

IB MHL Complex Numbers Page 1

Polar Form of a Complex Number

We have looked at how to express a complex number in terms of Cartesian
coordinates (x, y) for x+iy.

We can also describe a complex number using the modulus and argument (the
length of the complex number, r, and the angle, 𝜃, of its position vector to the x
axis).
This is expressed as (r, 𝜃 ). Label the diagram below showing r, 𝜃 and the
coordinates of P

What is the x coordinate in this diagram?

What is the y coordinate in this diagram?

How do we write a complex number in polar form from Cartesian form?

Is there a short form to write this?

(x,y) P
Re
Im
r
𝜃


Practice
1. Write in polar form:

1) 2i
2) -3
3) 1-i

2. Write in Cartesian form

2cis (
!
!
)
3 cis 𝜋
√2𝑐𝑖𝑠(−
𝜋
4
)


Properties of CIS 𝜽

When we learn about the compound angle identities we will be able to prove the
following results:

Can you explain the last result?

r1cis 𝜃 x r2cis 𝛽 = r1r2cis (𝜃+ 𝛽)

𝑟!cis 𝜃
𝑟!cisβ
=
𝑟!
𝑟!
𝑐𝑖𝑠(𝜃 − 𝛽)

𝑐𝑖𝑠𝜃 = 𝑐𝑖𝑠(𝜃 + 𝑘2𝜋)

Lesson 3 argument polar form of a complex number

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Lesson 3 argument polar form of a complex number