Post on 27-Mar-2015
Trigonometric (Polar) Form of
Complex Numbers
How is it Different?In a rectangular system, you
go left or right and up or down.
In a trigonometric or polar system, you have a direction to travel and a distance to travel in that direction.
2 2z i 2 cos45 sin 45
(2,45)
z i
Polar form
RealAxis
Imaginary Axis
Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane.
z a bi
b
a
z
2 2z a b
1tanb
a
The angle formed from the real axis and a line from the origin to (a, b) is called the argument of z, with requirement that 0 < 2.
modified for quadrant and so that it is between 0 and 2
z a bi The absolute value or modulus of z denoted by z is the distance from the origin to the point (a, b).
Trigonometric Form of a Complex Number
2 2The modulus is r a b
1
The argument can be found
by using tan adjusting for
correct quadrant if necessary
b
a
b
a
cos sinz r i
Note: You may use any other trig functions and their relationships to the
right triangle as well as tangent.
r
RealAxis
Imaginary Axis
r
�
Plot the complex number and then convert to trigonometric form: iz 3
Find the modulus r
1
3
2413 22r
3
1tan 1 but in Quad II
6
5
6
5sin
6
5cos2
iz
Find the argument
It is easy to convert from trigonometric to rectangular form because you just work the trig functions and distribute the r through.
i 3
6
5sin
6
5cos2
iz
i
2
1
2
32
2
3 2
1
If asked to plot the point and it is in trigonometric form, you would plot the angle and radius.
2
6
5 Notice that is the same as plotting
3 i 31
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7
Graphing Utility:Write the complex numberin standard form a + bi.
3 33.75 cos sin4 4
i
3 33.75 cos sin4 4
i
2.652 2.652i
[2nd] [decimal point]
Multiplying Complex NumbersTo multiply complex numbers in
rectangular form, you would FOIL and convert i2 into –1.
To multiply complex numbers in trig form, you simply multiply the rs and
and the thetas.
2
a bi c di
ac adi bci dbi
ac adi bci db
ac db ad bc i
1 1 1 2 2 2
1 2 1 2 1 2
cos sin cos sin
cos sin
r i r i
r r i
The formulas are scarier than they are.
use sum formula for sinuse sum formula for cos
Replace i 2 with -1 and group real terms and then imaginary terms
irr 2121212121 sincoscossinsinsincoscos
Must FOIL these
221121 sincossincos iirr
21 2 1 2 2 1 1 2 1 2cos cos sin cos sin cos sin sinr r i i i
Let's try multiplying two complex numbers in trigonometric form together.
1111 sincos irz 2222 sincos irz
1 2 1 1 1 2 2 2cos sin cos sinz z r i r i
212121 sincos irr
Look at where we started and where we ended up and see if you can make a statement as to what happens to the r 's and the 's when you multiply two complex numbers.
Multiply the Moduli and Add the Arguments
Example
1 2
1
2
2 3 2 4 cos30 sin 30
3 2 3 2 6 cos45 sin 45
multiply z z
Where z i i
z i i
1 2
2
2 3 2 3 2 3 2
6 6 6 6 6 2 6 2
6 6 6 6 6 2 6 2
6 6 6 2 6 6 6 2
z z
i i
i i i
i i
i
Rectangular form Trig form
1 2
4 cos30 sin 30 6 cos45 sin 45
4 6 cos 30 45 sin 30 45
24 cos75 sin 75
z z
i i
i
i
2 2
6 6 6 2 6 6 6 2
216 72 12 72 216 72 12 72
576 24
r
r
r
1 6 6 6 2tan 75
6 6 6 2
Dividing Complex Numbers
In rectangular form, you rationalize using the complex
conjugate.
2
2 2 2
2 2
2 2 2 2
a bi
c dia bi c di
c di c di
ac adi bci bdi
c d iac adi bci bd
c dac bd bc ad
ic d c d
In trig form, you just divide the rs and subtract the theta.
1 1 1
2 2 2
11 2 1 2
2
cos sin
cos sin
cos sin
r i
r i
ri
r
Then numbers.complex twobe
sincos and sincosLet 22221111 irzirz
zz
rr
i1
2
1
21 2 1 2 cos sin
then,0 If 2 z
(This says to multiply two complex numbers in polar form, multiply the moduli and add the arguments)
(This says to divide two complex numbers in polar form, divide the moduli and subtract the arguments)
21212121 sincos irrzz
wzzw
iwiz
(b) (a) :find
,120sin120cos6 and 40sin40cos4 If
4 cos 40 sin 40 6 cos120 sin120zw i i
12040sin12040cos64 i
24 cos160 sin160i
multiply the moduli add the arguments (the i sine term will have same argument)
If you want the answer in rectangular coordinates simply compute the trig functions and multiply the 24 through.
i34202.093969.024
i21.855.22
zw
i
i
4 40 40
6 120 120
cos sin
cos sin
46
40 120 40 120cos sin i
23
80 80cos sin i
23
280 280cos sin i
divide the moduli subtract the arguments
In polar form we want an angle between 0 and 360° so add 360° to the -80°
In rectangular coordinates:
ii 66.012.09848.01736.03
2
Example
1
2
1
2
3 2 3 2 6 cos45 sin 45
2 3 2 4 cos30 sin 30
zdivide
z
Where z i i
z i i
Rectangular form
2
2
3 2 3 2
2 3 2
3 2 3 2 2 3 2
2 3 2 2 3 2
6 6 6 2 6 6 6 2
12 4
6 6 6 2 6 6 6 2
12 4
6 6 6 2 6 6 6 2
16 16
i
i
i i
i i
i i i
i
i i
i
Trig form
6 cos45 sin 45
4 cos30 sin 30
6cos 45 30 sin 45 30
43
cos15 sin152
i
i
i
i
2 2
6 6 6 2 6 6 6 2
16 16
216 72 12 72 216 72 12 72
256
576 9 3
256 4 2
r
r
r
1
6 6 6 2
16tan 156 6 6 2
16
Powers of Complex NumbersThis is horrible in rectangular
form.
...
na bi
a bi a bi a bi a bi
The best way to expand one of these is using Pascal’s
triangle and binomial expansion.
You’d need to use an i-chart to simplify.
It’s much nicer in trig form. You just raise the r to the power and multiply theta by the exponent.
cos sin
cos sinn n
z r i
z r n i n
3 3
3
5 cos20 sin 20
5 cos3 20 sin 3 20
125 cos60 sin 60
Example
z i
z i
z i
Roots of Complex Numbers
• There will be as many answers as the index of the root you are looking for– Square root = 2 answers– Cube root = 3 answers, etc.
• Answers will be spaced symmetrically around the circle– You divide a full circle by the number of
answers to find out how far apart they are
General Process
1. Problem must be in trig form
2. Take the nth root of n. All answers have the same value for n.
3. Divide theta by n to find the first angle.
4. Divide a full circle by n to find out how much you add to theta to get to each subsequent answer.
The formula
cos sin
360 360 2 2cos sin cos sinn n n
z r i
k k k kz r i or r i
n n n n
k starts at 0 and goes up to n-1
This is easier than it looks.
Example Find the 4th root of 81 cos80 sin80z i
1. Find the 4th root of 81 4 81 3r
2. Divide theta by 4 to get the first angle.
8020
4
3. Divide a full circle (360) by 4 to find out how far apart
the answers are.
36090 between answers
4
4. List the 4 answers.
• The only thing that changes is the angle.
• The number of answers equals the number of roots.
1
2
3
4
3 cos20 sin 20
3 cos 20 90 sin 20 90 3 cos110 sin110
3 cos 110 90 sin 110 90 3 cos200 sin 200
3 cos 200 90 sin 200 90 3 cos290 sin 290
z i
z i i
z i i
z i i