1. Trigonometric Form of a Complex Number
Digital Lesson
Trigonometric Form of a Complex Number

2. Definition: Complex Plane
In the complex plane, every complex number corresponds to a point.
Example:
Imaginary axis
Real axis 2 4
– 2
Plot the points 3 + 4i and –2 – 2i in the complex plane.
(3, 4) or
3 + 4i
(– 2, – 2) or
– 2 – 2i
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Definition: Complex Plane

3. Definition: Absolute Value
The absolute value of the complex number z = a + bi is the distance between the origin (0, 0) and the point (a, b).
Example:
Plot z = 3 + 6i and find its absolute value.
Imaginary axis
Real axis 4
– 2
– 4 2 6 8
z = 3 + 6i
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Definition: Absolute Value

4. Trigonometric Form of a Complex Number
To write a complex number a + bi in trigonometric form, let  be the angle from the positive real axis (measured counter clockwise) to the line segment connecting the origin to the point (a, b).
Imaginary axis
Real axis b r a
(a, b) 
a = r cos 
b = r sin 
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Trigonometric Form of a Complex Number

5. Definition: Trigonometric Form of a Complex Number
The trigonometric form of a complex number z = a + bi is given by
z = r(cos  + i sin )
where a = r cos , b = r sin ,
The number r is the modulus of z, and  is the argument of z.
Example:
modulus
argument
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Definition: Trigonometric Form of a Complex Number

6. Example: Trigonometric Form of a Complex Number
Write the complex number z = –7 + 4i in trigonometric form.
Imaginary axis
Real axis
z = –7 + 4i
150.26°
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Example: Trigonometric Form of a Complex Number

7. Example: Standard Form of a Complex Number in Radians
Write the complex number
in standard form a + bi.
Example:
Standard form
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Example: Standard Form of a Complex Number in Radians

8. Example: Standard Form of a Complex Number in Degrees
Write the complex number
in standard form a + bi.
Example:
Standard form
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Example: Standard Form of a Complex Number in Degrees

9. Graphing Utility: Standard Form of a Complex Number
Write the complex number
in standard form a + bi.
[2nd] [decimal point]
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Graphing Utility: Standard Form of a Complex Number