Transcript Slide 1
Digital Lesson
Trigonometric Form of
a Complex Number
In the complex plane, every complex number
corresponds to a point.
Example:
Plot the points 3 + 4i and
–2 – 2i in the complex plane.
Imaginary axis
4
(3, 4)
or
3 + 4i
2
Real
axis
–2
(– 2, – 2)
or
– 2 – 2i
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2
–2
2
The absolute value of the complex number z = a + bi is
the distance between the origin (0, 0) and the point (a, b).
| a bi | a b
2
2
Example:
Plot z = 3 + 6i and find its absolute value.
Imaginary axis
z 3 6
2
8
6
z = 3 + 6i
9 36
4
–4
2
3 5 units
–2
4
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2
Real
axis
45
3 5
3
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
a = r cos
b = r sin
(a, b)
r a 2 b2
r
b
a bi (r cos ) (r sin )i
a
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Real
axis
4
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 , r a2 b2 , and tan b .
a
The number r is the modulus of z, and is the
argument of z.
Example:
z 1 cos i sin
2
3
3
modulus
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argument
5
Example:
Write the complex number z = –7 + 4i in trigonometric
form.
z r 7 4i
Imaginary axis
(7) 2 4 2 65
z = –7 + 4i
z 65
tan b 4
a
7
tan 1 4 29.74
7
180 29.74 150.26
150.26°
Real
axis
z r cos θ i sin θ
65(cos150.26 i sin150.26)
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6
Example:
Write the complex number 3.75 cos 3 i sin 3
4
4
in standard form a + bi.
cos 3 2
4
2
sin 3 2
4
2
2
2
z 3.75
i
2
2
15 2 15 2 i Standard form
8
8
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7
Example:
Write the complex number 3 cos330 i sin 330
2
in standard form a + bi.
cos330 3
2
sin 330 1
2
3 1
3
z
i
2 2 2
3 3 3 i Standard form
4
4
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8
Graphing Utility:
Write the complex number 3.75 cos 3 i sin 3
4
4
in standard form a + bi.
[2nd] [decimal point]
3.75 cos 3 i sin 3
4
4
2.652 2.652i
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9
Homework Assignment #50
Section 6.5 Sheet