-
Polar Coordinates
Jan 6­9:42 PM
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 - 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.
Jan 6­9:44 PM
1
Jan 6­11:26 PM
An angle rotated counterclockwise is positive.
An angle rotated clockwise is negative.
Jan 6­11:13 PM
2
Jan 6­11:31 PM
If "r" is negative, start at that negative value and then rotate
the require degrees
Jan 6­11:13 PM
3
Jan 6­11:32 PM
Jan 6­11:33 PM
4
http://www.analyzemath.com/polarcoordinates/plot_polar_coordinates.html
Jan 6­11:09 PM
homework; page 290 #1, 3
Dec 21­10:54 AM
5
Converting Between Rectangular Form and Polar Form
GOAL: To understand relationship between rectangular coordinates, (x, y)
and polar coordinates, (r, θ)
Jan 7­10:58 PM
Converting from Rectangular to Polar Form
Express (3,4) as a polar coordinate.
(3,4)
First - determine r.
How can we determine r?
(3,4)
r
Pythagorean Theorem
r
4
3
Now we need to determine
tan Θ = opp adj
5
4
3
So the rectangular coordinate (3,4) would be (5, 53
)in polar form.
Jan 7­11:04 PM
6
(3,4) = (5, 53 )
Jan 10­4:36 PM
Summarizing: To convert from rectangular to
polar form:
1. Find Θ. Tan Θ = y ∴ Θ = tan-1 y
x
x
2. Find r. Recall r = √ x2 + y2
3. Polar coordinates: (r, Θ)
Jan 10­8:16 PM
7
Quadrants
2
1
3
4
Jan 10­5:32 PM
Express ( -4, -4) in polar form.
­4
­4
(­4,­4)
BUT
o
α = 45
is not the whole rotation angle.
= + 180
= (45 + 180)
= 225
So (­4,­4) in polar form in (5.66, 225 )
Jan 10­4:00 PM
8
(-4,-4) in rectangular form is (5.66, 225 ) in polar form.
(5.66, 225 )
Jan 6­10:26 PM
What about points in the other two quadrants?
Express the (-5,12) in polar form.
(­5,12)
r
Now to find the angle
tan α = 12 ­5
= 180 ­ 67.4 = 112.6
(­5, 12 ) is ( 13, 112.6 ) in polar form.
Jan 6­10:26 PM
9
Express (8,-15) in polar form.
In which quadrant would (8, -15) be found?
4th
First find r
(8,-15)
x=8
y = -15
Now find
= 360 - 62 = 298
(8,-15) = (17, 298 )
Jan 7­11:03 PM
To Summarize:
1. If in Quad #1: θ = α
2. If in Quad #2: θ = 180o - α
3. If in Quad #3: θ = 180o + α
4. If in Quad #4: θ = 360o - α
Jan 10­8:26 PM
10
Converting from Polar Form to Rectangular Form
(4, 30 )
4
To find x
y
30
x
To find y
So ( 4, 30 ) in polar form is (3.46, 2) in rectangular form.
first quadrant
Jan 10­5:51 PM
Converting from Polar Form to Rectangular Form
(4, 150 )
(-3.64, 2)
Does this make sense?
Jan 7­11:03 PM
11
Recall that "a + bi" can be written as an
ordered pair: (a,b). "a" corresponds to "x"
and "b" corresponds to "y".
So it is with the polar coordinates:
a = rcosΘ ,
b = rsinΘ
a + bi = rcosΘ + rsinΘ(i)
= r(cosΘ +isinΘ)
= rcisΘ (pronounced "ciss")
where cisΘ = (cosΘ +isinΘ)
Jan 10­9:39 PM
To Summarize:
From Polar to Rectangular:
1. Find x: x = r(cosΘ)
OR a = r(cosΘ)
(5, 53.13o) = (3, 4)
- complex
2. Find y: y = r(sinΘ)
OR b= r(sinΘ)
- complex
3. The rectangular coordinates: (x, y) OR (a,b)
In complex form: a+bi
Jan 10­8:40 PM
12
Polar Form of a Complex Number
r cos
+ 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
4 cis 120
Jan 6­10:26 PM
Convert the complex number -6 + 8i into polar form.
Find r
Now find
In the Argand plane (-6+8i is in quadrant 2)
2nd quadrant
-6 + 8i would be 10 cis 127 in polar form
Jan 10­7:13 PM
13
Convert
a=
to rectangular form.
b=
y = -2.5
a=x
b=y
-4.3 - 2.5i
Jan 10­7:23 PM
Homework: page 295, #14c and d, 15 a, b
19, a,b,e f
21 a and b i, ii, iii
more review: 24 a, c
25
Jan 10­7:59 PM
14
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15
Jan 11­10:04 AM
16