 # Math 112 Elementary Functions

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Math 112 Elementary Functions
Chapter 7 – Applications of Trigonometry Section 3 Complex Numbers: Trigonometric Form

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. 1 y x x is a positive real number y is a negative real number

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)

Graphing Complex Numbers
Imaginary Axis -4 2 Real Axis All real numbers, a = a+0i, lie on the real axis at (a, 0).

Graphing Complex Numbers
Imaginary Axis All imaginary numbers, bi = 0+bi, lie on the imaginary axis at (0, b). 2i Real Axis -4i

Graphing Complex Numbers
Imaginary Axis 2 + 3i -4 + i Real Axis 3 – 2i -3 - 4i All other numbers, a+bi, are located at the point (a,b).

|x| = distance from the origin
Absolute Value Real Numbers: |x| = distance from the origin

|a + bi| = distance from the origin
Absolute Value Complex Numbers: |a + bi| = distance from the origin a + bi a b Note that if b = 0, then this reduces to an equivalent definition for the absolute value of a real number.

Trigonometric Form of a Complex Number
a + bi r a b Therefore, a + bi = r (cos + i sin) Note: As a standard,  is to be the smallest positive number possible.

Trigonometric Form of a Complex Number
Example: – 3i Steps for finding the trig form of a + bi. r = |a + bi|  is determined by … cos  = a / r sin  = b / r

Trigonometric Form of a Complex Number – Determining 
a + bi = r cis  r = |a+bi| 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.

Trigonometric Form of Real and Imaginary Numbers (examples)
Real/Imaginary Number Complex Form Trig w/ Degrees Radians 0 + 0i 0 cis 0° 0 cis 0 2 2 + 0i 2 cis 0° 2 cis 0 -5 -5 + 0i 5 cis 180° 2 cis  3i 0 + 3i 3 cis 90° 3 cis (/2) -4i 0 – 4i 4 cis 270° 4 cis (3/2)

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  i

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!

Multiplication of Complex Numbers (Standard Form)

Multiplication of Complex Numbers (Trigonometric Form)

Division of Complex Numbers (Standard Form)

Division of Complex Numbers (Trigonometric Form)

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

Powers of Complex Numbers (Trigonometric Form)
DeMoivre’s Theorem (r cis )n = rn cis (n)

Roots of Complex Numbers
An nth root of a number (a+bi) is any solution to the equation … xn = a+bi

Roots of Complex Numbers
Examples The two 2nd roots of 9 are … 3 and -3, because: = and (-3)2 = 9 The two 2nd roots of -25 are … 5i and -5i, because: (5i)2 = and (-5i)2 = -25 The two 2nd roots of 16i are … 22 + 22i and 2 - 22i because (22 + 22i)2 = 16i and (-22 - 22i)2 = 16i

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

Roots of Complex Numbers
In general, there are always … n “nth roots” of any complex number

Roots of Complex Numbers
One more example … Using DeMoivre’s Theorem Let k = 0, 1, & 2 NOTE: If you let k = 3, you get 2cis385 which is equivalent to 2cis25.

Roots of Complex Numbers
The n nth roots of the complex number r(cos  + i sin ) are …

Roots of Complex Numbers
The n nth roots of the complex number r cis  are … or

Summary of (r cis ) w/ r = 1
} Does this remind you of something?

Euler’s Formula Therefore, the complex number … r = |a + bi|
Note:  must be expressed in radians. Therefore, the complex number … r = |a + bi| cos  = a/r sin  = b/r

Results of Euler’s Formula
This gives a relationship between the 4 most common constants in mathematics!

Results of Euler’s Formula
ii is a real number!

Results of Euler’s Formula