Download Trig form of Complex Numbers

Trig form of Complex Numbers
Objective: Be able to operate with complex numbers,
and be able to convert complex numbers into Trig
Form and vise versa.
TS: Examining information from more than one view
point.
Warm-Up: Do the following operations:
a)(3 + 5i) + (4 – 2i)
b) (3 + 5i)(4 – 2i)
Complex Numbers
a+bi where i=√-1 (i2 = -1)
Remember:
They can be graphed on complex plane
Imaginary axis
3 + 2i
1
Real axis
1
Absolute Value
• Absolute value is the _________________
So |a + bi| =
Imaginary
Find |-3 + 4i|
1
|-2 – 6i|
1
Real
Trig form of Complex Numbers
To effectively work with powers and roots, it is helpful to use trig to
express imaginary numbers.
If θ is the angle formed to point (a, b)
then
Imaginary
Thus a
r
θ
b
Real
a = r cos θ & b = r sin θ
+ bi = (r cos θ) + (r sin θ) I
= r cis θ
where r =
a
a b
2
2
(r is called the modulus)
and
b
tan  
a
(θ is called the argument)
Switching Between Forms
Write each in trigonometric form
1) 2 + 2i
2) -1 – √3i
Switching Between Forms
Write each in trigonometric form
3) 2.5(√3 – i)
4) 7
Switching Between Forms
Write each in trigonometric form
5) 1 + 2i
Switching Between Forms
Write each in standard form
1) 5(cos135° + i sin135°) 2) ¾ cis 330°
Switching Between Forms
Write each in standard form
3)
4)



1.5  cos  i sin 
2
2

5
4cis
6
You Try
Represent 4 – 4√3i graphically, and find the
trigonometric form of a number. Also find
the absolute value of it.
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answers
5
Trig form  8cis
3
Imaginary
4  4 3i  8
Real
4 – 4√3i
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Question