1.2
Complex Number &
Trigonometric identities
TMA1101 Calculus
For lecture session TC102 only
Faculty of Computing and Informatics
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Lecturer information
Chin Wen Cheong
FCI BR4012
Email: [email protected]
Phone: 03-83125249
Website: http://pesona.mmu.edu.my/~wcchin
Faculty of Computing and Informatics
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Number
3
COMPLEX NUMBER
4
Operation: sum and difference
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Operation: product
6
Operation: division
7
conjugates
8
Principal square root
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Patterns of i
5
4
i  1
i  i  i  (1)i  i
i 2  1
i 6  i 4  i 2  (1)(1)  1
3
2
i  i  i  (1)i  i
4
2
2
i  i  i  (1)(1)  1
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4
3
i  i  i  (1)(i )  i
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4
4
i  i  i  (1)(1)  1
The powers recycle through each multiple of 4.
i
4k
1
10
Examples
Write using i notation.
a.
 25 
25   1  5 i
 32 
32   1  16  2   1  4 2  i  4 i 2
b.
c.


  121   121   1   11i
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Trigonometric functions
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Trigonometric functions
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Trigonometric functions
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Trigonometric functions
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Trigonometric functions
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Trigonometric functions
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
Divide by cos2:
Divide by sin2:
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Trigonometric Identities
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Trigonometric Identities
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Trigonometric Identities
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Complex number: Polar form
 
Suppose the polar coordinates of the point are  2, 
 4
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25
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Polar form
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Polar form
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Polar form
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More on polar form: multiplication
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More on polar form
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More on polar form: division
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More on polar form: division
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If z  4  cos 35  i sin 35  and w  2  cos 80  i sin 80  find
(a) zw
z
(b)
w
Leave answer in polar form.
(a) zw =  4  cos 35  i sin 35    2  cos80  i sin 80  
=  4  2  cos  35  80   i sin  35  80    8cos115  i sin115
z 4  cos 35  i sin 35 
4
(b)
=
= cos  35  80   i sin  35  80  
w 2  cos 80  i sin 80 
2
 2 cos  45   i sin  45    2  cos 45  i sin 45 
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Another form: Complex Exponential
Taylor series and Taylor Swift is different:
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Another form: Complex Exponential
Knowing:
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Complex Exponential
38
Complex Exponential
Trigonometric Identity
39
Application in calculus
Preliminary algebra before integration
2
40
Complex Exponential
Preliminary algebra before integration
41