Post on 03-Jan-2016
9.6 Polar Coordinates
Digital Lesson
One way to give someone directions is to tell them to go three blocks East and five blocks South.
Another way to give directions is to point and say “Go a half mile in that direction.”
Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.
Initial ray
r A polar coordinate pair
determines the location of a point.
,r
9.6 Polar Coordinates
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The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).
= directed angle Polar axis
r = directed dista
nce
OPole (Origin)
The polar axis is the ray constructed from O.
Each point P in the plane can be assigned polar coordinates (r, ).
P = (r, )
r is the directed distance from O to P.
is the directed angle (counterclockwise) from the polar axis to OP.
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a) The point lies two units from the pole on the
terminal side of the angle
( , ) 2,3
r
.3
3
2,3
2
32
1 2 3 0
Plotting Points
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33,4
34
2
32
1 2 3 0
3 units from the pole
Plotting Points
b) Plot the point 33,
4
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There are many ways to represent the point 2, .3
2
32
1 2 3 0
2,3
52,3
2 42,2 2,,3 3 3
52,3
additional ways
to represent the
point 2,3
( , ) , 2r r n
( , ) , (2 1)r r n
• Plot the polar point , then find one additional representation.
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3,4
2
32
1 2 3 0
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(r, )(x, y)
Polex
y
(Origin)
y
r
x
The relationship between rectangular and polar coordinates is as follows.
The point (x, y) lies on a circle of radius r, therefore,
r2 = x2 + y2.
tan yx
cos xr
sin yr
Definitions of trigonometric functions
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Coordinate Conversion
cosx r cos xr
siny r sin yr
2 2 2r x y tan yx
(Pythagorean Identity)
Example:Convert the point into rectangular coordinates. 4,
3
1cos co3
24 s 42
x r
3sin sin 4 23 2
4 3y r
, 2, 2 3x y
x = r cos(θ)
y = r sin(θ)
r2 = x2 + y2
tan(θ) = y/x
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Example:
Convert the point into rectangular coordinates. 2,
Example:
Convert the point into rectangular coordinates. 3,6
Example:
Convert the point into rectangular coordinates. 4.5,1.3
x = r cos(θ)
y = r sin(θ)
2,0
3 3,
2 2
1.204,. 4.336
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Example:
Convert the point (1,1) into polar coordinates.
, 1,1x y
1tan 11
yx
4
2 2 2 21 1 2r x y
set of polar coordinates is ( , ) 2, .4
One r
5Another set is ( , ) 2, .4
r
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Example:
Convert the point (0,2) into polar coordinates.
Example:
Convert the point (-3,4) into polar coordinates.
r2 = x2 + y2
tan(θ) = y/x
2,2
5, 2.214
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Equation Conversion
x = r cos(θ)
y = r sin(θ)
r2 = x2 + y2
tan(θ) = y/x
To convert a rectangular equation to polar form, use:
To convert a polar equation to rectangular form, use:
Equation Conversion
Example For the rectangular equation 3x + 2y = 4,
(a) convert to a polar equation,
(b) use a graphing calculator to graph the rectangular equation, and
(c) use a graphing calculator to graph the polar equation for 0° 360°.
Solution
(a) Let x = r cos and y = r sin to get
3 cos 2 sin 4 or4
.3cos 2sin
rr r
Converting a Rectangular Equation to Polar Form
(b) Solve the rectangular equation for y to get
(c)
.223
xy
Converting a Polar Equation to Rectangular Form
Example For the polar equation
(a) convert to a rectangular equation,
(b) use a graphing calculator to graph the polar equation for 0 2, and
(c) use a graphing calculator to graph the rectangular equation.
Solution: Multiply both sides by the denominator.
41 sin ,r
sin 4r r 2 2 4x y y 2
22 2 4x y y
2 2 216 8x y y y 2 8 2x y
Converting a Polar Equation to Rectangular Form
(b) The figure showsa graph with polarcoordinates.
(c) Solving x2 = –8(y – 2)for y, we obtain
.2 281 xy
Homework Day 1
9.6 pg. 680:
1, 5, 9, 13, 17, 21, 25, 29, 33, 37-55 odds
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9.6 Polar CoordinatesDay 2
Digital Lesson
Answer equation conversion problems at end of lesson.
HWQ 3/25
Find 2 different sets of polar coordinates
for the rectangular point:
Exact values only.
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3, 1
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9.6 Day 2 - More Equation Conversion
x = r cos(θ)
y = r sin(θ)
r2 = x2 + y2
tan(θ) = y/x
To convert a rectangular equation to polar form, use:
To convert a polar equation to rectangular form, use:
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Converting Equations
Ex: Convert the rectangular equation to polar form: 2y x
sec tanr
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Ex: Convert the rectangular equation to polar form: 2 2 16x y
4r
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3 6 2 0x y Ex: Convert the rectangular equation to
polar form:
2
6sin 3cosr
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Equation Conversion
Convert the polar equation into a rectangular equation.
4sinr
4sinr 2 4 sinr r Multiply each side by r.
2 2 4x y y Substitute rectangular coordinates.
22 2 4x y Equation of a circle with center (0, 2) and radius of 2
Polar form
2 2 4 0x y y
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Example:
Convert the polar equation r = 2 to rectangular form.
2 2 4x y
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Example:
Convert the polar equation to rectangular form.
3
3y x
tan tan3
3y
x
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Example:
Convert the polar equation to rectangular form. secr
x = r cos(θ)
y = r sin(θ)
cos sec cosr
1x sec cosx
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Example:
Convert from rectangular to polar form:2 3x y
2cot cscr
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Example:
Convert the polar equation to rectangular form.
x = r cos(θ)
y = r sin(θ)
3cosr
223 9
2 4x y
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Example:
Convert the polar equation to rectangular form.
x = r cos(θ)
y = r sin(θ)
2 sin 2r
22 2 2x y xy
Homework Day 2
9.6 pg. 680
3, 7, 11, 15, 19, 23, 27, 31, 35, 57-77 odds
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HWQ 3/26
• Convert the rectangular equation to polar form:
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2 2 8 0x y y