PARAMETRIC EQUATIONS AND POLAR...
Transcript of PARAMETRIC EQUATIONS AND POLAR...
![Page 1: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/1.jpg)
PARAMETRIC EQUATIONS AND POLAR COORDINATES
10
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A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates.
PARAMETRIC EQUATIONS & POLAR COORDINATES
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Usually, we use Cartesian coordinates, which are directed distances from twoperpendicular axes.
PARAMETRIC EQUATIONS & POLAR COORDINATES
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Here, we describe a coordinate systemintroduced by Newton, called the polarcoordinate system.
It is more convenient for many purposes.
PARAMETRIC EQUATIONS & POLAR COORDINATES
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10.3Polar Coordinates
In this section, we will learn:
How to represent points in polar coordinates.
PARAMETRIC EQUATIONS & POLAR COORDINATES
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POLE
We choose a point in the plane that is called the pole (or origin) and is labeled O.
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POLAR AXIS
Then, we draw a ray (half-line) starting at O called the polar axis.
This axis is usually drawn horizontally to the rightcorresponding to the positive x-axis in Cartesiancoordinates.
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ANOTHER POINT
If P is any other point in the plane, let:
r be the distance from O to P. θ be the angle (usually measured in radians)
between the polar axis and the line OP.
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POLAR COORDINATES
P is represented by the ordered pair (r, θ).
r, θ are called polar coordinates of P.
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POLAR COORDINATES
We use the convention that an angle is:
Positive—if measured in the counterclockwisedirection from the polar axis.
Negative—if measured in the clockwise direction from the polar axis.
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If P = O, then r = 0, and we agree that (0, θ) represents the pole for any value of θ.
POLAR COORDINATES
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We extend the meaning of polar coordinates (r, θ) to the case in which r is negative—as follows.
POLAR COORDINATES
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POLAR COORDINATES
We agree that, as shown, the points (–r, θ)and (r, θ) lie on the same line through O and at the same distance | r | from O, but on opposite sides of O.
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POLAR COORDINATES
If r > 0, the point (r, θ) lies in the samequadrant as θ.
If r < 0, it lies in the quadrant on the opposite side of the pole.
Notice that (–r, θ) represents the same point as (r, θ + π).
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POLAR COORDINATES
Plot the points whose polar coordinates are given.
a. (1, 5π/4)b. (2, 3π)c. (2, –2π/3)d. (–3, 3π/4)
Example 1
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POLAR COORDINATES
The point (1, 5π/4) is plotted here.Example 1 a
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The point (2, 3π) is plotted.Example 1 bPOLAR COORDINATES
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POLAR COORDINATES
The point (2, –2π/3) is plotted.Example 1 c
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POLAR COORDINATES
The point (–3, 3π/4) is plotted.
It is is located three units from the pole in the fourth quadrant.
This is because the angle 3π/4 is inthe second quadrant and r = -3 is negative.
Example 1 d
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CARTESIAN VS. POLAR COORDINATES
In the Cartesian coordinate system, everypoint has only one representation.
However, in the polar coordinate system,each point has many representations.
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CARTESIAN VS. POLAR COORDINATES
For instance, the point (1, 5π/4) in Example 1 a could be written as: (1, –3π/4), (1, 13π/4), or (–1, π/4).
![Page 22: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/22.jpg)
CARTESIAN & POLAR COORDINATES
In fact, as a complete counterclockwiserotation is given by an angle 2π, the pointrepresented by polar coordinates (r, θ) is also represented by
(r, θ + 2nπ) and (-r, θ + (2n + 1)π)
where n is any integer.
![Page 23: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/23.jpg)
CARTESIAN & POLAR COORDINATES
The connection between polar and Cartesiancoordinates can be seen here.
The pole corresponds to the origin. The polar axis coincides with the positive x-axis.
![Page 24: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/24.jpg)
CARTESIAN & POLAR COORDINATES
If the point P has Cartesian coordinates (x, y) and polar coordinates (r, θ), then, from the figure, we have: cos sinx y
r rθ θ= =
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cos
sin
x r
y r
θ
θ
=
=
Equations 1
Therefore,CARTESIAN & POLAR COORDS.
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CARTESIAN & POLAR COORDS.
Although Equations 1 were deduced from the figure (which illustrates the case wherer > 0 and 0 < θ < π/2), these equations are valid for all values of r and θ.
See the generaldefinition of sin θand cos θin Appendix D.
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CARTESIAN & POLAR COORDS.
Equations 1 allow us to find the Cartesian coordinates of a point when the polar coordinates are known.
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CARTESIAN & POLAR COORDS.
To find r and θ when x and y are known,we use the equations
These can be deduced from Equations 1 or simply read from the figure.
2 2 2 tan yr x yx
θ= + =
Equations 2
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CARTESIAN & POLAR COORDS.
Convert the point (2, π/3) from polar to Cartesian coordinates.
Since r = 2 and θ = π/3, Equations 1 give:
Thus, the point is (1, ) in Cartesian coordinates.
Example 2
1cos 2cos 2 13 2
3sin 2sin 2. 33 2
x r
y r
πθ
πθ
= = = ⋅ =
= = = =
3
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CARTESIAN & POLAR COORDS.
Represent the point with Cartesiancoordinates (1, –1) in terms of polarcoordinates.
Example 3
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CARTESIAN & POLAR COORDS.
If we choose r to be positive, then Equations 2 give:
As the point (1, –1) lies in the fourth quadrant, we can choose θ = –π/4 or θ = 7π/4.
Example 3
2 2 2 21 ( 1) 2
tan 1
r x yyx
θ
= + = + − =
= = −
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CARTESIAN & POLAR COORDS.
Thus, one possible answer is: ( , –π/4)
Another possible answer is:
( , 7π/4)
Example 3
2
2
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CARTESIAN & POLAR COORDS.
Equations 2 do not uniquely determine θwhen x and y are given.
This is because, as θ increases through the interval 0 ≤ θ ≤ 2π, each value of tan θ occurs twice.
Note
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CARTESIAN & POLAR COORDS.
So, in converting from Cartesian to polar coordinates, it’s not good enough just to find rand θ that satisfy Equations 2.
As in Example 3, we must choose θ so that the point (r, θ) lies in the correct quadrant.
Note
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POLAR CURVES
The graph of a polar equation r = f(θ) [or, more generally, F(r, θ) = 0] consists of all points that have at least one polarrepresentation (r, θ), whose coordinatessatisfy the equation.
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POLAR CURVES
What curve is represented by the polar equation r = 2 ?
The curve consists of all points (r, θ) with r = 2.
r represents the distance from the point to the pole.
Example 4
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POLAR CURVES
Thus, the curve r = 2 represents the circle with center O and radius 2.
Example 4
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POLAR CURVES
In general, the equation r = a represents a circle O with center and radius |a|.
Example 4
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POLAR CURVES
Sketch the polar curve θ = 1.
This curve consists of all points (r, θ) such that the polar angle θ is 1 radian.
Example 5
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POLAR CURVES
It is the straight line that passes through O and makes an angle of 1 radian with the polaraxis.
Example 5
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POLAR CURVES
Notice that:
The points (r, 1) on the line with r > 0 are in the first quadrant.
The points (r, 1) on the line with r < 0 are in the third quadrant.
Example 5
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POLAR CURVES
a. Sketch the curve with polar equationr = 2 cos θ.
b. Find a Cartesian equation for this curve.
Example 6
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POLAR CURVES
First, we find the values of r for some convenient values of θ.
Example 6 a
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POLAR CURVES
We plot the corresponding points (r, θ).
Then, we join these points to sketch the curve—as follows.
Example 6 a
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The curve appears to be a circle.Example 6 aPOLAR CURVES
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POLAR CURVES
We have used only values of θ between 0 and π—since, if we let θ increase beyond π, we obtain the same points again.
Example 6 a
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POLAR CURVES
To convert the given equation to a Cartesian equation, we use Equations 1 and 2.
From x = r cos θ, we have cos θ = x/r.
So, the equation r = 2 cos θ becomes r = 2x/r.
This gives:2x = r2 = x2 + y2 or x2 + y2 – 2x = 0
Example 6 b
![Page 48: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/48.jpg)
POLAR CURVES
Completing the square, we obtain:
(x – 1)2 + y2 = 1
The equation is of a circle with center (1, 0) and radius 1.
Example 6 b
![Page 49: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/49.jpg)
POLAR CURVES
The figure shows a geometrical illustrationthat the circle in Example 6 has the equationr = 2 cos θ.
The angle OPQ is a right angle, and so r/2 = cos θ.
Why is OPQa right angle?
![Page 50: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/50.jpg)
POLAR CURVES
Sketch the curve r = 1 + sin θ.
Here, we do not plot points as in Example 6.
Rather, we first sketch the graph of r = 1 + sin θin Cartesian coordinates by shifting the sine curveup one unit—as follows.
Example 7
![Page 51: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/51.jpg)
POLAR CURVES
This enables us to read at a glance the values of r that correspond to increasingvalues of θ.
Example 7
![Page 52: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/52.jpg)
POLAR CURVES
For instance, we see that, as θ increases from 0 to π/2, r (the distance from O)increases from 1 to 2.
Example 7
![Page 53: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/53.jpg)
POLAR CURVES
So, we sketch the corresponding part of the polar curve.
Example 7
![Page 54: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/54.jpg)
POLAR CURVES
As θ increases from π/2 to π, the figure shows that r decreases from 2 to 1.
Example 7
![Page 55: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/55.jpg)
POLAR CURVES
So, we sketch the next part of the curve.
Example 7
![Page 56: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/56.jpg)
POLAR CURVES
As θ increases from to π to 3π/2,r decreases from 1 to 0, as shown.
Example 7
![Page 57: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/57.jpg)
POLAR CURVES
Finally, as θ increases from 3π/2 to 2π, r increases from 0 to 1, as shown.
Example 7
![Page 58: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/58.jpg)
POLAR CURVES
If we let θ increase beyond 2π or decrease beyond 0, we would simply retrace our path.
Example 7
![Page 59: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/59.jpg)
POLAR CURVES
Putting together the various parts of the curve, we sketch the complete curve—as shown next.
Example 7
![Page 60: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/60.jpg)
CARDIOID
It is called a cardioid—because it’s shaped like a heart.
Example 7
![Page 61: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/61.jpg)
POLAR CURVES
Sketch the curve r = cos 2θ.
As in Example 7, we first sketch r = cos 2θ,0 ≤ θ ≤2π, in Cartesian coordinates.
Example 8
![Page 62: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/62.jpg)
POLAR CURVES
As θ increases from 0 to π/4, the figure shows that r decreases from 1 to 0.
Example 8
![Page 63: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/63.jpg)
POLAR CURVES
So, we draw the corresponding portion of the polar curve(indicated by ).1
Example 8
![Page 64: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/64.jpg)
POLAR CURVES
As θ increases from π/4 to π/2, r goes from 0 to –1.
This means that the distance from O increases from 0 to 1.
Example 8
![Page 65: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/65.jpg)
POLAR CURVES
However, instead of being in the first quadrant,this portion of the polar curve (indicated by ) lies on the opposite side of the pole in the third quadrant.
2
Example 8
![Page 66: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/66.jpg)
POLAR CURVES
The rest of the curve is drawn in a similar fashion.
The arrows and numbersindicate the order in which the portions are traced out.
Example 8
![Page 67: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/67.jpg)
POLAR CURVES
The resulting curvehas four loops and is called a four-leaved rose.
Example 8
![Page 68: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/68.jpg)
SYMMETRY
When we sketch polar curves, it is sometimes helpful to take advantage of symmetry.
![Page 69: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/69.jpg)
RULES
The following three rules are explained by figures.
![Page 70: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/70.jpg)
RULE 1
If a polar equation is unchanged when θis replaced by –θ, the curve is symmetricabout the polar axis.
![Page 71: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/71.jpg)
RULE 2
If the equation is unchanged when r is replaced by –r, or when θ is replaced byθ + π, the curve is symmetric about the pole.
This means that the curve remainsunchanged if we rotate it through 180° about the origin.
![Page 72: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/72.jpg)
RULE 3
If the equation is unchanged when θ isreplaced by π – θ, the curve is symmetricabout the vertical line θ = π/2.
![Page 73: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/73.jpg)
SYMMETRY
The curves sketched in Examples 6 and 8 are symmetric about the polar axis, since cos(–θ) = cos θ.
![Page 74: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/74.jpg)
SYMMETRY
The curves in Examples 7 and 8 aresymmetric about θ = π/2, becausesin(π – θ) = sin θ and cos 2(π – θ) = cos 2θ.
![Page 75: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/75.jpg)
SYMMETRY
The four-leaved rose is also symmetricabout the pole.
![Page 76: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/76.jpg)
SYMMETRY
These symmetry properties could have been used in sketching the curves.
![Page 77: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/77.jpg)
SYMMETRY
For instance, in Example 6, we need onlyhave plotted points for 0 ≤ θ ≤ π/2 and thenreflected about the polar axis to obtain the complete circle.
![Page 78: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/78.jpg)
TANGENTS TO POLAR CURVES
To find a tangent line to a polar curve r = f(θ), we regard θ as a parameter and write its parametric equations as:
x = r cos θ = f (θ) cos θ
y = r sin θ = f (θ) sin θ
![Page 79: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/79.jpg)
TANGENTS TO POLAR CURVES
Then, using the method for finding slopes of parametric curves (Equation 2 in Section 10.2) and the Product Rule, we have:
Equation 3
sin cos
cos sin
dy dr rdy d ddx drdx rd d
θ θθ θ
θ θθ θ
+= =
−
![Page 80: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/80.jpg)
TANGENTS TO POLAR CURVES
We locate horizontal tangents by finding the points where dy/dθ = 0 (provided that dx/dθ ≠ 0).
Likewise, we locate vertical tangents at the points where dx/dθ = 0 (provided that dy/dθ ≠ 0).
![Page 81: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/81.jpg)
TANGENTS TO POLAR CURVES
Notice that, if we are looking for tangent lines at the pole, then r = 0 and Equation 3 simplifies to:
tan if 0dy drdx d
θθ
= ≠
![Page 82: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/82.jpg)
TANGENTS TO POLAR CURVES
For instance, in Example 8, we found that r = cos 2θ = 0 when θ = π/4 or 3π/4.
This means that the lines θ = π/4 and θ = 3π/4(or y = x and y = –x) are tangent lines to r = cos 2θat the origin.
![Page 83: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/83.jpg)
TANGENTS TO POLAR CURVES
a. For the cardioid r = 1 + sin θ of Example 7, find the slope of the tangent line when θ = π/3.
b. Find the points on the cardioid where the tangent line is horizontal or vertical.
Example 9
![Page 84: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/84.jpg)
TANGENTS TO POLAR CURVES
Using Equation 3 with r = 1 + sin θ, we have:
2
sin cos
cos sin
cos sin (1 sin )coscos cos (1 sin )sin
cos (1 2sin ) cos (1 2sin )1 2sin sin (1 sin )(1 2sin )
dr rdy ddrdx rd
θ θθ
θ θθ
θ θ θ θθ θ θ θ
θ θ θ θθ θ θ θ
+=
−
+ +=
− +
+ += =
− − + −
Example 9
![Page 85: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/85.jpg)
TANGENTS TO POLAR CURVES
The slope of the tangent at the point whereθ = π/3 is:
Example 9 a
3
12
cos( / 3)(1 2sin( / 3))(1 sin( / 3))(1 2sin( / 3)
(1 3)(1 3 / 2)(1 3)
1 3 1 3 1(2 3)(1 3) 1 3
dydx θ π
π ππ π=
+=
+ −
+=
+ −
+ += = = −
+ − − −
![Page 86: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/86.jpg)
TANGENTS TO POLAR CURVES
Observe that:Example 9 b
dydθ
= cosθ(1+ 2sinθ) = 0 whenθ =π2
,3π2
,7π6
,11π6
dxdθ
= (1+ sinθ)(1− 2sinθ) = 0 whenθ =3π2
,π6
,5π6
![Page 87: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/87.jpg)
TANGENTS TO POLAR CURVES
Hence, there are horizontal tangents at the points
(2, π/2), (½, 7π/6), (½, 11π/6)and vertical tangents at
(3/2, π/6), (3/2, 5π/6)
When θ = 3π/2, both dy/dθ and dx/dθ are 0. So, we must be careful.
Example 9 b
![Page 88: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/88.jpg)
TANGENTS TO POLAR CURVES
Using l’Hospital’s Rule, we have:
(3 / 2)
(3 / 2) (3 / 2)
(3 / 2)
(3 / 2)
lim
1 2sin coslim lim1 2sin 1 sin
1 coslim3 1 sin1 sinlim3 cos
dydxθ π
θ π θ π
θ π
θ π
θ θθ θθθθθ
−
− −
−
−
→
→ →
→
→
+ = − +
= −+−
= − = ∞
Example 9 b
![Page 89: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/89.jpg)
TANGENTS TO POLAR CURVES
By symmetry,
(3 / 2)lim dy
dxθ π +→= −∞
Example 9 b
![Page 90: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/90.jpg)
TANGENTS TO POLAR CURVES
Thus, there is a vertical tangent line at the pole.
Example 9 b
![Page 91: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/91.jpg)
TANGENTS TO POLAR CURVES
Instead of having to remember Equation 3, we could employ the method used to derive it.
For instance, in Example 9, we could have written:
x = r cos θ = (1 + sin θ) cos θ = cos θ + ½ sin 2θ
y = r sin θ = (1 + sin θ) sin θ = sin θ + sin2θ
Note
![Page 92: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/92.jpg)
TANGENTS TO POLAR CURVES
Then, we would have
which is equivalent to our previous expression.
Note
/ cos 2sin cos/ sin cos 2
cos sin 2sin cos 2
dy dy ddx dx d
θ θ θ θθ θ θ
θ θθ θ
+= =
− ++
=− +
![Page 93: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/93.jpg)
GRAPHING POLAR CURVES
It’s useful to be able to sketch simple polar curves by hand.
![Page 94: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/94.jpg)
GRAPHING POLAR CURVES
However, we need to use a graphing calculator or computer when faced with curves as complicated as shown.
![Page 95: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/95.jpg)
GRAPHING POLAR CURVES WITH GRAPHING DEVICES
Some graphing devices have commands that enable us to graph polar curves directly.
With other machines, we need to convert to parametric equations first.
![Page 96: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/96.jpg)
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In this case, we take the polar equation r = f(θ) and write its parametric equations as:
x = r cos θ = f(θ) cos θ
y = r sin θ = f(θ) sin θ
Some machines require that the parameter be called t rather than θ.
![Page 97: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/97.jpg)
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Graph the curve r = sin(8θ / 5).
Let’s assume that our graphing device doesn’t have a built-in polar graphing command.
Example 10
![Page 98: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/98.jpg)
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In this case, we need to work with the corresponding parametric equations, which are:
In any case, we need to determine the domain for θ.
Example 10
cos sin(8 / 5)cos
sin sin(8 / 5)sin
x r
y r
θ θ θ
θ θ θ
= =
= =
![Page 99: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/99.jpg)
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So, we ask ourselves:
How many complete rotations are required until the curve starts to repeat itself ?
Example 10
![Page 100: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/100.jpg)
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If the answer is n, then
So, we require that 16nπ/5 be an even multiple of π.
8( 2 ) 8 16sin sin5 5 5
8sin5
n nθ π θ π
θ
+ = +
=
Example 10
![Page 101: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/101.jpg)
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This will first occur when n = 5.
Hence, we will graph the entire curve if we specify that 0 ≤ θ ≤ 10π.
Example 10
![Page 102: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/102.jpg)
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Switching from θ to t, we have the equations
sin(8 / 5)cossin(8 / 5)sin
0 10
x t ty t t
t π
==≤ ≤
Example 10
![Page 103: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/103.jpg)
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This is the resulting curve.
Notice that this rose has 16 loops.
Example 10
![Page 104: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/104.jpg)
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Investigate the family of polar curves given by r = 1 + c sin θ.
How does the shape change as c changes?
These curves are called limaçons—after a French word for snail, because of the shape of the curves for certain values of c.
Example 11
![Page 105: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/105.jpg)
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The figures show computer-drawn graphs for various values of c.
Example 11
![Page 106: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/106.jpg)
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For c > 1, there is a loop that decreases in size as decreases.
Example 11
![Page 107: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/107.jpg)
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When c = 1, the loop disappears and the curve becomes the cardioid that we sketched in Example 7.
Example 11
![Page 108: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/108.jpg)
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For c between 1 and ½, the cardioid’s cusp is smoothed out and becomes a “dimple.”
Example 11
![Page 109: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/109.jpg)
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When c decreases from ½ to 0, the limaçon is shaped like an oval.
Example 11
![Page 110: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/110.jpg)
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This oval becomes more circular as c → 0.
When c = 0, the curve is just the circle r = 1.
Example 11
![Page 111: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/111.jpg)
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The remaining parts show that, as cbecomes negative, the shapes change in reverse order.
Example 11
![Page 112: PARAMETRIC EQUATIONS AND POLAR COORDINATESrfrith.uaa.alaska.edu/M201/Chapter10/Chap10_Sec3.pdfCARTESIAN & POLAR COORDS. So, in converting from Cartesian to polar coordinates, it’s](https://reader034.fdocuments.net/reader034/viewer/2022052501/60805e246b78f80bb973afb4/html5/thumbnails/112.jpg)
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In fact, these curves are reflections about the horizontal axis of the corresponding curves with positive c.
Example 11