Math 112 Elementary Functions

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Math 112Elementary Functions
Chapter 7 Applications of Trigonometry

• Section 3
• Complex Numbers Trigonometric Form

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Graphing Complex Numbers

• How do you graph a real number?
• Use a number line.
• The point corresponding to a real number
  represents the directed distance from 0.

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Graphing Complex Numbers

• General form of a complex number
• a bi
• a ? R and b ? R
• i ?-1

Therefore, a complex number is essentially an
ordered pair! (a, b)
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Graphing Complex Numbers
Imaginary Axis
Real Axis
All real numbers, a a0i, lie on the real axis
at (a, 0).
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Graphing Complex Numbers
Imaginary Axis
All imaginary numbers, bi 0bi, lie on the
imaginary axis at (0, b).
Real Axis
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Graphing Complex Numbers
Imaginary Axis
Real Axis
All other numbers, abi, are located at the point
(a,b).
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Absolute Value

• Real Numbers
• x distance from the origin

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Absolute Value

• Complex Numbers
• a bi distance from the origin

Note that if b 0, then this reduces to an
equivalent definition for the absolute value of a
real number.
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Trigonometric Form of aComplex Number
?
Therefore, a bi r (cos? i
sin?)
Note As a standard, ? is to be the smallest
positive number possible.
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Trigonometric Form of aComplex Number
Example 2 3i

• Steps for finding the trig form of a bi.
• r a bi
• ? is determined by
• cos ? a / r
• sin ? b / r

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Trigonometric Form of aComplex Number
Determining ?

• a bi r cis ?
• r abi cos ? a/r
  sin ? b/r

• Using cos ? a/r
• Q1 ? cos-1(a/r)
• Q2 ? cos-1(a/r)
• Q3 ? 360 - cos-1(a/r)
• Q4 ? 360 - cos-1(a/r)

• Using sin ? b/r
• Q1 ? sin-1(b/r)
• Q2 ? 180 - sin-1(b/r)
• Q3 ? 180 - sin-1(b/r)
• Q4 ? 360 sin-1(b/r)

For Radians, replace 180 with ? and 360 with 2?.
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Trigonometric Form of Real and Imaginary Numbers
(examples)
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Converting the Trigonometric Form to Standard Form

• r cis ?
• r (cos ? i sin ?)
• (r cos ?) (r sin ?) i
• Example 4 cis 30º
• (4 cos 30º) (4 sin 30º)i
• 4(?3)/2 4(1/2)i
• 2?3 2i
• ? 3.46 2i

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Arithmetic with Complex Numbers

• Addition Subtraction
• Standard form is very easy Trig. form is ugly!
• Multiplication Division
• Standard form is ugly.Trig. form is easy!
• Exponentiation Roots
• Standard form is very ugly.Trig. form is very
  easy!

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Multiplication of Complex Numbers(Standard Form)
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Multiplication of Complex Numbers(Trigonometric
Form)
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Division of Complex Numbers(Standard Form)
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Division of Complex Numbers(Trigonometric Form)
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Powers of Complex Numbers(Trigonometric Form)

• r cis ?2
• (r cis ?) (r cis ?)
• r2 cis(? ?)
• r2 cis 2?

• r cis ?3
• (r cis ?)2 (r cis ?)
• r2 cis(2?) (r cis ?)
• r3 cis 3?

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Powers of Complex Numbers(Trigonometric Form)

• DeMoivres Theorem
• (r cis ?)n rn cis (n?)

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Roots of Complex Numbers

• An nth root of a number (abi) is any solution to
  the equation
• xn abi

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Roots of Complex Numbers

• Examples
• The two 2nd roots of 9 are
• 3 and -3, because 32 9 and
  (-3)2 9
• The two 2nd roots of -25 are
• 5i and -5i, because (5i)2 -25 and
  (-5i)2 -25
• The two 2nd roots of 16i are
• 2?2 2?2i and -2?2 - 2?2i
• because (2?2 2?2i)2 16i and
  (-2?2 - 2?2i)2 16i

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Roots of Complex Numbers

• Example
• Find all of the 4th roots of 16.
• x4 16
• x4 16 0
• (x2 4)(x2 4) 0
• (x 2i)(x 2i)(x 2)(x 2) 0
• x 2i or 2

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Roots of Complex Numbers

• In general, there are always
• n nth roots of any complex number

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Roots of Complex Numbers

• One more example

Using DeMoivres Theorem
Let k 0, 1, 2
NOTE If you let k 3, you get 2cis385? which is
equivalent to 2cis25?.
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Roots of Complex Numbers
The n nth roots of the complex number r(cos ? i
sin ?) are
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Roots of Complex Numbers
The n nth roots of the complex number r cis ? are

or
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Summary of (r cis ?) w/ r 1
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Eulers Formula
Note ? must be expressed in radians.
Therefore, the complex number

• r a bi
• cos ? a/r
• sin ? b/r

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Results of Eulers Formula
This gives a relationship between the 4 most
common constants in mathematics!
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Results of Eulers Formula
ii is a real number!
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Results of Eulers Formula