1. 10.4 Trigonometric (Polar) Form of Complex Numbers
The Complex Plane and Vector Representations
Call the horizontal axis the real axis and the vertical axis the imaginary axis. Now complex numbers can be graphed in this complex plane.
The sum of two complex numbers can be represented graphically by the vector that is the resultant of the sum of vectors corresponding to the two numbers.

2. 10.4 Expressing the Sum of Complex Numbers Graphically
Example Find the sum of 6 – 2i and –4 – 3i. Graph
both complex numbers and their resultant.
Solution (6 – 2i) + (–4 – 3i) = 2 – 5i

3. 10.4 Trigonometric (Polar) Form
The graph shows the complex number x + yi that corresponds to the vector OP.
Relationship Among x, y, r, and 

4. 10.4 Trigonometric (Polar) Form
Substituting x = r cos  and y = r sin  into x + yi gives
Trigonometric or Polar Form of a Complex Number
The expression
r(cos  + i sin  )
is called the trigonometric form (or polar form) of the complex number x + yi.

5. 10.4 Trigonometric (Polar) Form
Notation:
cos  + i sin  is sometimes written cis  . Using this notation, r(cos  + i sin  ) is written r cis .
The number r is called the modulus or absolute value of the complex number x + yi.
Angle  is called the argument of the complex number x + yi.

6. 10.4 Converting from Trigonometric Form to Rectangular Form
Example Express 2(cos 300º + i sin 300º) in
rectangular form.
Analytic Solution
Graphing Calculator Solution

7. 10.4 Converting from Rectangular to Trigonometric Form
Sketch a graph of the number x + yi in the complex plane.
Find r by using the equation
Find  by using the equation tan  = y/x, x  0, choosing the quadrant indicated in Step 1.

8. 10.4 Converting from Rectangular to Trigonometric Form
Example Write each complex number in trigonometric
form.
Solution
Start by sketching the graph
of in the complex
plane.
Then find r.

9. 10.4 Converting from Rectangular to Trigonometric Form
Now find .
Therefore, in polar form,
 is in quadrant II and tan  = the reference angle in quadrant II is

10. 10.4 Converting from Rectangular to Trigonometric Form
(b)
From the graph,  = 270º. In trigonometric form,
different way to determine .

11. 10.4 Products of Complex Numbers in Trigonometric Form
Multiplying complex numbers in rectangular form.
Multiplying complex numbers in trigonometric form.

12. 10.4 Products of Complex Numbers in Trigonometric Form
Product Theorem
If are any two complex numbers, then
In compact form, this is written

13. 10.4 Using the Product Theorem
Example Find the product of 3(cos 45º + i sin 45º) and
2(cos 135º + i sin 135º).
Solution

14. 10.4 Quotients of Complex Numbers in Trigonometric Form
The rectangular form of the quotient of two complex numbers.
The polar form of the quotient of two complex numbers.

15. 10.4 Quotients of Complex Numbers in Trigonometric Form
Quotient Theorem
If r1(cos 1 + i sin 1) and r2(cos 2 + i sin 2) are complex numbers, where r2(cos 2 + i sin 2)  0, then
In compact form, this is written

16. 10.4 Using the Quotient Theorem
Example Find the quotient Write the result in rectangular form.
Solution