Math 112Elementary Functions

Section 3

Complex Numbers: Trigonometric Form

Chapter 7 – Applications of Trigonometry

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.

0 1y xx 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 NumbersImaginary

Axis

Real Axis

All real numbers, a = a+0i, lie on the real axis at (a, 0).

2-4

Graphing Complex NumbersImaginary

Axis

Real Axis

All imaginary numbers, bi = 0+bi, lie on the imaginary axis at (0, b).

2i

-4i

Graphing Complex NumbersImaginary

Axis

Real Axis

All other numbers, a+bi, are located at the point (a,b).

2 + 3i

-4 + i

3 – 2i

-3 - 4i

Absolute Value

Real Numbers:|x| = distance from the origin

0 if

0 if

xx

xxx

2xx

Absolute Value

Complex Numbers:|a + bi| = distance from the origin

22 babia

Note that if b = 0, then this reduces to an equivalent definition for the absolute value of a real number.

a + bi

ab

Trigonometric Form of aComplex Number

22 babiar

a + bi

a

b

r cos cos rar

a

sin sin rbr

b

Therefore,

a + bi = r (cos + i sin)

Note: As a standard, is to be the smallest positive number possible.

Trigonometric Form of aComplex Number

cisr

irbia

)sin(cos

1332 22 r

304 cis 135.3 cis 1332 Therefore, i

Steps for finding the trig form of a + bi.

• r = |a + bi|

• is determined by …cos = a / rsin = b / r

Example: 2 – 3i

5698.13

2cos 1

5698.13

3sin 1

Trigonometric Form of aComplex 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

Trig w/

Radians

0 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= 23 + 2i 3.46 + 2i

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)

ibcadbdacdicbia

Multiplication of Complex Numbers(Trigonometric Form)

)sin(cos sincos isir

sincoscossinsinsincoscos iirs

sincos irs

cis cis cis rssr

Division of Complex Numbers(Standard Form)

22

)()(

dc

iadbcbdac

dic

bia

Division of Complex Numbers(Trigonometric Form)

)sin(cos

)sin(cos

is

ir

)sin(cos

)sin(cos

)sin(cos

)sin(cos

i

i

is

ir

)sin(cos

sincoscossinsinsincoscos22

ii

s

r

sincos is

r

cis cis

cis

s

r

s

r

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: 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 … 22 + 22i and -22 - 22i

because (22 + 22i)2 = 16i and (-22 - 22i)2 = 16i

Roots of Complex Numbers

Example:

Find all of the 4Find all of the 4thth roots of 16. roots of 16.

x4 = 16x4 – 16 = 0 (x2 + 4)(x2 – 4) = 0 (x + 2i)(x – 2i)(x + 2)(x – 2) = 0x = ±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 …

31

36075 8 kcis 3 57 8 cis 31

75 8 cis

265 2 ,145 2 ,25 2 cisciscis

3

36075 2

kcis

kcis 12025 2

Let k = 0, 1, & 2

Using DeMoivre’s

Theorem

NOTE: If you let k = 3, you get 2cis385 which is equivalent to 2cis25.

Roots of Complex Numbers

1 ..., 4, 3, 2, 1, 0, where

360sin

360cos

1

nk

nk

ni

nk

nr n

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

Roots of Complex Numbers

1 ..., 4, 3, 2, 1, 0, where nk

n

kcisr n

3601

The n nth roots of the complex number r cis are …

n

kcisr n

21

or

Summary of (r cis ) w/ r = 1

BAcisBcisAcis

BAcisBcis

Acis

nAcisAcis n }

Does this remind youof something?

abba

bab

a

baba

xx

xx

x

xxx

Euler’s Formula

sincos iei

)sin(cos ir bia

Therefore, the complex number …

ire

cisr

Note: must be expressed in radians.

r = |a + bi|

cos = a/r

sin = b/r

Results of Euler’s Formula

01 1 ii ee ie

This gives a relationship between the 4 most

common constants in mathematics!

Results of Euler’s Formula

i

ii

ie

2

2

ie

iie

2

...2078795763.0 ii

i

ii is a real number!

Results of Euler’s Formula

2

ixix ee xcos

2

ixix ee xsin