Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q...
Transcript of Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q...
![Page 1: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/1.jpg)
There are many ways to mark points in the plane or in 3-dim space for purposes of navigation. In the familiar rectangular coordinate system, a point is chosen as the origin and a perpendicular set of lines is drawn through that point, one horizontal and one vertical. A unit of Length is chosen, and every point is given a pair of coordinates (x, y) indicating its distance, horizontally and vertically from the origin.
The choice of origin, axes, and length is completely arbitrary.
Polar Coordinates
![Page 2: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/2.jpg)
(0,0)1
1
2 3-1-2-3
-1
-2
-3
2
3 (2, 3)
![Page 3: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/3.jpg)
An alternative method of assigning coordinates is shown below.
Here we draw a straight line from the origin to the point. We then assign to the point the angle θ that the line makes with the positive x axis, and the distance r from the origin to the point.
Some examples are:
θ
r(r,θ )
![Page 4: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/4.jpg)
(2, π/4)
(1, π/4)
(2, 0)
(1, π/2)
![Page 5: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/5.jpg)
(3, π/6)(1, 3π/4)
(2, −π/3)(2, −π/2)
![Page 6: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/6.jpg)
The relationship between rectangular and polar coordinates is shown in the diagram below.
It is summarized by the equations:
x = rcos(θ)
y = rsin(θ)
2 2r x y= +
1arctan tany yx x
θ −= =
The angle θ is taken to be between −π and π.
θr y
x
![Page 7: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/7.jpg)
Problem. The polar coordinates of several points are given below. Find the rectangular coordinates of each of those points.
(a) (3, π/4) (b) (1, −π/3) (c) (4, 3π/4) (d) (1, −π/6)
Solution.(a)
(b)
(c)
(d)
3 3cos( /4) ;2
x π= = 3 3sin( /4) ;2
y π= = 3 3,2 2
1 cos( /3) ;2
x π= − = 3 sin( /3) ;2
y π= − =−1 3,2 2
−
4 4cos(3 /4) ;2
x π= =− 4 4sin(3 /4) ;2
y π= =4 4,2 2
−
3 cos( /6) ;2
x π= − =1 sin( /6) ;2
y π= − =−3 1,
2 2
−
![Page 8: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/8.jpg)
Problem. The rectangular coordinates of several points are given below. Find the polar coordinates of each of those points.
(a) (5, 5) (b) (1, −√3) (c) (−3, 3√3) (d) (−2, −2)
Solution.
(a) 2 2 50 5 2;r x y= + = = 51 tan ;5 4
πθ −= = 5 2,4π
(5, 5)
![Page 9: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/9.jpg)
Solution.
(b) 2 2 2;r x y= + = ( )1 tan 3 ;3πθ −= − =− 2,
3π −
(1, −√3)
Problem. The rectangular coordinates of several points are given below. Find the polar coordinates of each of those points.
(a) (5, 5) (b) (1, −√3) (c) (−3, 3√3) (d) (−2, −2)
2
![Page 10: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/10.jpg)
Solution.
(c) 2 2 6;r x y= + = ( ) 21 tan 3 ;3πθ −= − =
26,3π
(− 3, 3√3)
Problem. The rectangular coordinates of several points are given below. Find the polar coordinates of each of those points.
(a) (5, 5) (b) (1, −√3) (c) (−3, 3√3) (d) (−2, −2)
6
![Page 11: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/11.jpg)
Solution.
(d) 2 2 8 2 2;r x y= + = = ( ) 31 tan 1 ;4πθ −= − =−
32 2,4π −
(−2, −2)
Problem. The rectangular coordinates of several points are given below. Find the polar coordinates of each of those points.
(a) (5, 5) (b) (1, −√3) (c) (−3, 3√3) (d) (−2, −2)
![Page 12: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/12.jpg)
We consider the problem of graphing functions of the form r = f(θ) in polar coordinates, where the angle is measured in radians. First we look at two simple cases
Plot the function r = c in polar coordinates, where c is a positive constant.
This is the case where c = 2.
![Page 13: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/13.jpg)
Plot the function θ = c in polar coordinates, where c is a constant.
θ = c radians
In trigonometric calculations, we usually require r to be positive. However, when graphing curves r = f(θ) in polar coordinates, we allow r to be negative (since f(θ) often is) and interpret this to mean measuring “backward” from the origin along the ray.
positive r
negative r
![Page 14: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/14.jpg)
We now consider the problem of graphing more complicated functions.
Plot the function r = sin(θ) in polar coordinates.
![Page 15: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/15.jpg)
There are two ways to see how this plot looks. One is to change from polar to rectangular coordinates. The equation r = sin(θ) can also be written as r2 = rsin(θ) , or x2 + y2 = y. By completing the square, we can write this as
1 12 24 4
x y y+ − + = or2 21 12
2 2x y + − =
![Page 16: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/16.jpg)
A second way is to note that as θ increases, the line connecting the origin to the corresponding point on the curve sweeps around counterclockwise like the hand of a clock. At each value of θ the curve is plotted on that line a distance f(θ) from the origin. Thus as the line revolves around the origin, the point on the curve slides up and down the line. Proceed as follows:
Locate the value of r forθ = 0. Then as θ increases to 90 degrees, the point slides in or out until it reaches the correctlocation of r corresponding to θ = π/2. This gives an idea of how the curve looks in the first quadrant.
Continue this process in each quadrant.
![Page 17: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/17.jpg)
For the function r = sin(θ) , the initial position of the point isr = 0 when θ = 0. At θ = π/2, the point is at a distance sin(π/2) = 1 along the “clock hand”. Thus as the hand sweeps out that 90 degree angle, the point moves up the hand from 0 to 1. The result is clearly the picture shown below, at least approximately.
![Page 18: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/18.jpg)
As the clock hand moves another 90 degrees to π, the point must move back along the clock hand to sin(π) = 0. Thus the picture continues as shown below.
At this point we have the entire circle. In the next two quadrants, r is negative. Thus the point moves to the negative part of the hand. The circle is painted out again in this way, and the process then repeats forever.
![Page 19: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/19.jpg)
Now let us look at the similar curve r = 2cos(θ). When θ = 0, r = 2. At θ = π/2, r = 2cos(π/2) = 0. Therefore, as the angle of the hand moves from 0 to π/2, the point on the curve moves down the hand from 2 to 0. This is shown below.
![Page 20: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/20.jpg)
As the angle of the hand moves from π/2 to π, the cosine becomes negative, so the point is plotted back through the origin, a negative distance. Thus r continues down the hand from 0 to −2.
![Page 21: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/21.jpg)
This figure appears to be a circle as well. To verify this we can translate into rectangular coordinates. If r = 2cos(θ), then r2 = 2rcos(θ), or x2 + y2 = 2x. As before, we can complete the square to obtain the equation
2 22 1 1x x y− + + =or
2 2( 1) 1x y− + =
This is clearly the equation of a circle with radius 1 and center (1, 0).
![Page 22: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/22.jpg)
Graph the curve r = 1 + cos(θ) in polar coordinates.
In this case, the change to rectangular coordinates would not yield a familiar curve. Thus we proceed in the straightforward way. At θ = 0, r is 2 and at θ = π/2, r is 1. This is illustrated below.
![Page 23: Polar Coordinates - College of Computing & Informatics angle q is taken to be between −p and p. q r y x. Problem. The polar coordinates of several points are given below.](https://reader033.fdocuments.in/reader033/viewer/2022051801/5adabde07f8b9ae1768d8e28/html5/thumbnails/23.jpg)
Finally, as the angle of the hand moves from 3π/2 to 2π , the cosine goes from 0 to 1, and so r moves up the hand from 1 to 2.
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The result is called a cardioid because of its heart shape. Here are some other examples.
r = 1 − cos(θ) r = 1 + sin(θ)
r = 1 − sin(θ)