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Transcript
Graphing Points on Polar
Paper
POLE: the common centre, O, of a series of
concentric circles
POLAR AXIS: a horizontal ray drawn from the
pole in a positive direction.
Polar Coordinates
POLAR COORDINATES - consist of an ordered
pair, P
where
is the distance from the
pole to point P, and
is the measure of the
angle formed by the polar axis and the terminal
arm, OP.
Plot on polar paper.
(+,+)
An angle rotated counterclockwise is positive.
An angle rotated clockwise is negative.
(+,-)
(2,-60)
(-,+)
(-,-)
http://www.analyzemath.com/polarcoordinates/plot_polar_coordinates.html
Converting from Rectangular to Polar Form
Express (3,4) as a polar coordinate.
(3,4)
First - determine r.
How can we determine r?
Converting Between Rectangular Form and Polar Form
(3,4)
r
To understand the relationship between rectangular coordinates, (x, y) and
polar coordinates, (r, θ)
Pythagorean Theorem
r
4
3
Now we need to determine
5
4
3
So the rectangular coordinate (3,4) would be (5, 53
)in polar form.
(3,4) = (5, 53 )
Quadrants
2
1
3
4
Express ( -4, -4) in polar form.
­4
(-4,-4) in rectangular form is (5.66, 225 ) in polar form.
­4
(­4,­4)
BUT
is not the whole rotation angle.
It is the reference or related angle.
(5.66, 225 )
= + 180
= (45 + 180)
= 225
So (­4,­4) in polar form is (5.66, 225 )
Express (8,-15) in polar form.
In which quadrant would (8, -15) be found?
What about points in the other two quadrants?
Express the (-5,12) in polar form.
First find r
(8,-15)
x=8
y = -15
(­5,12)
2nd quadrant
4th
r
Now find
Now to find the angle
= 360 - 62 = 298
= 180 ­ 67.4 = 112.6
(­5, 12 ) is ( 13, 112.6 ) in polar form.
(8,-15) = (17, 298 )
Converting from Polar Form to Rectangular Form
(4, 30 )
To find x
4
y
Converting from Polar Form to Rectangular Form
30
x
To find y
(4, 150 )
x= ­3.46
(-3.46, 2)
Does this make sense?
So ( 4, 30 ) in polar form is (3.46, 2) in rectangular form.
first quadrant
Polar Form of a Complex Number
Converting between Rectangular (complex #) and Polar
Complex number
Consider the Argand plane which is
used to plot complex numbers of
the form a + bi.
The point is plotted as (a,b) as
opposed to (x,y)
Therefore
a=
where
r cos
a + bi
a = r cos
b = r sin
+ ri sin
This can be factored as
r(cos
+ i sin
)
This is usually shortened to r cis
4 cos 120 + 4 i sin 120
b=
4 cis 120
polar form of a complex number
Convert the complex number -6 + 8i into polar form.
Convert 4-3i into polar form
Find r
4th quadrant
ALWAYS CONSIDER THE QUADRANT
Now find
In the Argand plane (-6+8i is in quadrant 2)
Find
2nd quadrant
4-3i would be in the 4th quadrant.
360-37=323
-6 + 8i would be 10 cis 127 in polar form
4-3i would be 5 cis 323 in polar form.
Convert
to rectangular form.
Convert 7cis 120 into rectangular form
b = -2.5
-3.5+6.1i
-4.3 - 2.5i
Multiplying Complex Numbers in Polar Form
Example 1:
convert to polar form first
REVIEW
Convert into rectangular form
a=5cos 300
a= 2.5
b=5sin300
b= ­4.33
(1+i)(-2+2i)
-2+2i -2i +2i2
-2-2
-4
Quadrant 1
Quadrant 2
1+i-2 + 2i
2.5­4.3i
In polar form
4cis180
Convert into polar form
4cis 180
a+bi
10cos 233.1 +10isin233.1
Multiply the radii, or moduli (the r values), and add the angles,
or arguments.
What is 4cis 180 in rectangular form?
a=4cos 180 b=4sin 180
a=-4 b=0
a+bi=-4
This checks
EXAMPLE 2:
On Handout
(√2cis45)(2√2cis135)
illustrated on previous pages
10√3+10i
illustrated on previous page
3+11i
(√13cis56.3)(√10cis18.4) √130cis74.7
Multiply the radii, or moduli (the r values), and add the angles, or arguments.
20cis30
a=20cos 30 b=20sin 30
a=17.32 b=10
17.32+10i
This checks
Multiply the radii, or moduli (the r values), and add the angles, or arguments.
DIVIDING COMPLEX NUMBERS IN POLAR FORM
Work it out in rectangular form.
Work it out in polar form.
To multiply complex numbers in polar form:
Quadrant 2
r1r2cis (α+β)
Multiply the radii, or moduli (the r values), and add the angles, or arguments.
NOTE
When dividing complex numbers in polar form,divide the moduli and subtract the arguments.
Now convert back to rectangular:
This checks with -0.5i
r1 cis(α-β)
r2
√2cis45
2√2cis135
-0.2i
0.9+0.7i
When dividing complex numbers in polar form,divide the moduli and subtract the
arguments.
√13cis56.3
√10cis18.4
1.14cis37.9
When dividing complex numbers in polar form,divide the moduli and subtract the arguments.
#6 page 302
Complete for homework
To multiply out using complex numbers is time consuming.
Is there another way?
Evaluate by converting to polar form: #6 page 302
(1+ i)8
4√2 cis 135
2cis 300
10cis 210
Is there a faster way to solve this question?
Now evaluate using the rules for multiplying polar coordinates
Now convert back to rectangular complex form
2√2cis 315
34.64 ­20i
Rectangular Form
(1+ i)8
Now let's apply DeMoivre's Theorem to this
question.
DeMoivre's Theorem
note complex number must be in polar form
(rcisθ)n = rn cis nθ
Example 1:
(4cis225)3
4 cis(3x225)
64cis 675 675-360
64cis 315
First convert to polar form
r = √2 θ=45
(√2 cis 45)8
Apply DeMoivre's Theorem
3
Example 2
(3cis50)4
34 cis (4x50)
81 cis 200
(√2)8 cis ( 8 x 45)
16 cis 360
Convert back to rectangular complex
form
a = 16 cos 360 b = 16 sin 360
a = 16 b=0
16
(1+i)8 = 16
Powers of Complex Numbers
Example 2:
Example 1
(-8+15i)2
First convert to polar form (2nd quadrant)
(3+4i)4
Convert to polar form
(17 cis 118.07)2
(5cis 53.13)4
Apply DeMoivre's Theorem
2
17 cis(2x118.07)
289cis236.14
Now convert back to rectangular
complex form
a = 289 cos 236.14
a=-161.02
(-8+15i)2
b = 289sin 236.14
b=-239.99
= -161.02 -239.99i
THE END
Apply DeMoivre's Theorem
54 cis(4x53.13)
625 cis 212.52
Convert back to complex form
a=625cos212.52b=625sin 212.52
a=-526.94b=-336
(3+4i)4 = -526.94 - 336i