Download Ch 8 Notes - El Camino College

Math 120
Trigonometry Lecture Notes
Chapter 8
8.1 Complex Numbers
Imaginary Numbers:
i
= _____
i 2 = ________
Complex Numbers:
 Have a _________________ and an ________________________ part.
 a + bi
a and b are real numbers, and b ≠ 0
 Complex Conjugate: _______________________
When do we see Complex Numbers?
Quadratic Formula: Used to solve any quadratic equation of the form: ax2 + bx + c = 0
x=
− b ± b 2 − 4ac
2a
1) x2 + 27 = 0
2) 3x2 – 5x + 8 = 0
Arithmetic with Complex Numbers:
A) Adding & Subtracting: Add or Subtract the Real & Complex parts separately. (They never combine)
Subtract: (2 + 4i ) – (1 – 6i ) – (2 + 5i ) =
B) Multiplying & Dividing:
a) Change negative radicands to i BEFORE multiplying or dividing.
− 10 ⋅ − 14
7
1)
− 15 ⋅ − 21
1)
b) Multiplying: Use Distributive Property or FOIL, then simplify powers of i, then combine like terms.
(4 – 3i)
2) (5 + 2i ) (4 – 3i)
i
2)
3) (5 + 2i ) (5 – 2i)
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Math 120
Trigonometry Lecture Notes
Chapter 8
c) Dividing: We want to separate the real and imaginary parts of the number, so we can’t leave i in the
denominator. The Trick: Multiply the denominator by the complex conjugate. Simplify, then split into two
separate fractions.
1)
3 − 5i
1 +i
2)
Powers of i
To Find: to any power:
a) Divide the power by 4
b) Look at the decimal
i) 0.25 means
ii) 0.5 means
iii) 0.75 means
iv) no decimal means
i = ________
i 2 = ________
i 3 = ________
i 4 = ________
Find i 74
7 + 4i
8 − 6i
i -85
Application: Complex numbers are used to describe current, I, voltage, E, and impedance, Z (the opposition to
current). These three quantities are related by the equation: E = IZ. Find E if I = 3 + 4i and Z = 8 + 6i
CA: 8.1 35, 45, 55, 61, 69
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Math 120
Trigonometry Lecture Notes
Chapter 8
8.2 Trigonometric Form of Complex Numbers
z is used to represent complex numbers. z can be represented in several different forms:
•
z = a + bi
•
z = r (cos θ + i sin θ) = r cis θ
•
z = r eiθ
•
arg(z) = θ, the smallest positive angle θ from
the POSITIVE REAL axis to the graph of z.
(Rectangular Form)
(Trigonometric or Polar Form)
(Exponential Form)
To convert from Rectangular to Trigonometric form and vice versa:
a) Graph the number on “Complex Plane”
b) Consider the x-axis to be the ____________ axis and the y-axis to be the ______________ axis.
c) Draw the corresponding triangle and solve.
Practice:
1) Convert from Rectangular to Polar:
a) 3 – 3i
b) 4 + 2i
2) Convert from Polar to Rectangular:
a) 2 (cos 120⁰ + i sin 120⁰)
b) 5 cis 230⁰
We can represent these as vectors & use vector addition (using components).
Find the sum and represent both complex numbers and their resultant in trigonometric form.
-7 - i and 5 - 2i
CA: 8.2 # 21, 29, 45, 49
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Math 120
Trigonometry Lecture Notes
Chapter 8
8.3 Product & Quotient Theorems:
Multiply Two Complex Numbers:
(1 + i 3 )(− 2
3 + 2i
)
Now Change to Polar Form & Multiply:
Product Theorem:
r1 (cos θ1 + i sin θ1) ∙ r2 (cos θ2 + i sin θ2) =
Quotient Theorem:
Divide:
1+ i 3
− 2 3 + 2i
Now Change to Polar Form:
Quotient Theorem:
r1 (cos θ1 + i sin θ1 )
=
r2 (cos θ 2 + i sin θ 2 )
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Math 120
Trigonometry Lecture Notes
Chapter 8
Examples:
1) Find the product in rectangular form:
a) [3(cos 120⁰ + i sin 120⁰)] [4(cos 270⁰ + i sin 270⁰)]
b) (4.5 cis 267⁰) (1.8cis 14⁰)
2) Find the quotient in rectangular form:
a)
15(cos 225 + i sin 225)
3 (cos 45 + i sin 45)
b)
−i
1+ i
What happens if we square a complex number in Polar Coordinates?
[r cis θ]2
General Form for De Moivre’s Theorem:
[r cis θ]n =
This formula makes raising any complex number to a power, very simple:
1. Change to Polar Form
2. Use De Moivre’s Thm.
3. Change back to Rectangular Cooordinates
Example:
(-1 + i)4 =
CA 8.3 #15, 25, 29
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