Polar Coordinates Objective: To look at a different way to plot points and create a graph.
Transcript of Polar Coordinates Objective: To look at a different way to plot points and create a graph.
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Polar Coordinates
Objective: To look at a different way to plot points and create a graph
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Polar Coordinates
• A polar coordinate system in a plane consists of a fixed point O, called the pole (or origin), and a ray emanating from the pole, called the polar axis. In such a coordinate system we can associate with each point P in the plane a pair of polar coordinates (r, ), where r is the distance from P to the pole and is an angle from the polar axis to the ray OP. The number r is called the radial coordinate of P and the number the angular coordinate (or polar angle).
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Polar Coordinates
• The points (6, 45o), (5, 120o), (3,225o), and (4, 330o) are plotted below. If P is the pole, then r = 0, but there is no clearly defined polar angle. We will agree that an arbitrary angle can be used in this case; that is, (0, ) are polar coordinates of the pole for all choices of .
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Polar Coordinates
• The polar coordinates of a point are not unique. For example, the polar coordinates (1, 315o), (1, -45o), and (1, 675o) all represent the same point. In general, if a point P has polar coordinates (r, ), then (r, + n360o) and (r, - n360o) are also polar coordinates of P for any nonnegative integer n. Thus, every point has infinitely many pairs of polar coordinates.
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Polar Coordinates
• As defined above, the radial coordinate r of a point P is nonnegative, since it represents the distance from P to the pole. However, it will be convenient to allow for negative values of r as well.
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Polar Coordinates
• To motivate an appropriate definition, consider the point P with polar coordinates (3, 225o). We can reach this point by rotating the polar axis through an angle of 225 and moving 3 units along the pole, or we can reach the point P by rotating the polar axis through an angle of 45 and then moving 3 units from the pole along the extension of the terminal side.
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Relationship Between Polar and Rectangular Coordinates
• Frequently, it will be useful to superimpose a rectangular xy-coordinate system on top of a polar coordinate system, making the positive x-axis coincide with the polar axis. If this is done, the every point P will have both rectangular coordinates (x, y) and polar coordinates (r, ).
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Relationship Between Polar and Rectangular Coordinates
• Looking at the figure, we can see that
leading to the relationship
• This is how you would change polar coordinates to rectangular coordinates.
r
xcos
r
ysin
cosrx sinry
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Relationship Between Polar and Rectangular Coordinates
• Looking at the figure, we can also see that
• This is how you would change rectangular coordinates to polar coordinates.
222 yxr x
ytan
x
y1tan
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Example 1
• Find the rectangular coordinates of the point P whose polar coordinates are (6, 2/3).
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Example 1
• Find the rectangular coordinates of the point P whose polar coordinates are (6, 2/3).
• Since r = 6 and = 2/3, we have
)cos(6 32x )sin(6 3
2y
3
)(6 21
x
x
33
)(6 23
y
y
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Example 2
• Find the polar coordinates of the point P whose rectangular coordinates are .
• Since x = -2 and y = , we have
• Since we are in the second quadrant,
4
16)32()2( 222
r
r
)32,2(
32
2321 )(tan
32
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Example 2
• Find the polar coordinates of the point P whose rectangular coordinates are .
• Since x = -2 and y = , we have
• Since we are in the second quadrant,
4
16)32()2( 222
r
r
)32,2(
32
2321 )(tan
32
norn 2,42,4 35
32
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Graphs in Polar Coordinates
• We will now consider the problem of graphing equations in r and , where is assumed to be measured in radians. In a rectangular coordinate system the graph of an equation in x and y consists of all points whose coordinates (x, y) satisfy the equation.
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Graphs in Polar Coordinates
• However, in a polar coordinate system, points have infinitely many different pairs of polar coordinates, so that a given point may have some polar coordinates that satisfy an equation and others that do not. Given an equation in r and , we define its graph in polar coordinates to consist of all points with at least one pair of coordinates (r, ) that satisfy the equation.
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Example 3
• Sketch the graphs of (a) r = 1 (b) /4
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Example 3
• Sketch the graphs of (a) r = 1 (b) /4
(a) For all values of , the point (1, ) is 1 unit away from the pole. Since is arbitrary, the graph is the circle of radius 1 centered at the pole.
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Example 3
• Sketch the graphs of (a) r = 1 (b) /4
(b) For all values of r, the point (r, /4) lies on a line that makes an angle of /4 with the polar axis. Positive values of r correspond to points on the line in the first quadrant and negative values of r to points on the line in the third quadrant.
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Graphs
• Equations r = f() that express r as a function of are especially important. One way to graph such an equation is to choose some typical values of , calculate the corresponding values of r, and then plot the resulting pairs (r, ) in a polar coordinate system.
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Example 4
• Sketch the graph of r = (> 0) in polar coordinates by plotting points.
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Example 4
• Sketch the graph of r = (> 0) in polar coordinates by plotting points.
• Observe that as increases, so does r; thus, the graph is a curve that spirals out from the pole as increases. A reasonably accurate sketch of the spiral can be obtained by plotting points that correspond to values of that are integer
multiples of /2, keeping in mind that the value of r is always equal to the value of .
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Example 5
• Sketch the graph of the equation r = sin in polar coordinates by plotting points.
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Example 5
• Sketch the graph of the equation r = sin in polar coordinates by plotting points.
• The table below shows the coordinates of points on the graph at increments of /6. Note that 13 points are listed but we plotted only 7.
Many points represent the same place on the graph.
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Example 5
• Sketch the graph of the equation r = sin in polar coordinates by plotting points.
• This is what the graph looks like in a rectangular r- coordinate system.
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Example 5
• Observe that the points appear to lie on a circle. We can confirm that this is so by expressing the polar equation r = sin in terms of x and y. To do this, we multiply the equation through by r to obtain
sin2 rr
yyx 22
412
212 )( yx
222 yxr sinry
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Graphs
• Just because an equation r = f() involves the variables r and does not mean that it has to be graphed in a polar coordinate system. When useful, this equation can also be graphed in a rectangular coordinate system. For example, the graphs below are both the graph of r = sin.
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Example 6
• Sketch the graph of r = cos2 in polar coordinates.
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Example 6
• Sketch the graph of r = cos2 in polar coordinates.• We will use the graph of r = cos2 in rectangular
coordinates to visualize the graph in polar coord.
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Families of Rose Curves
• Equations of the following form are called rose curves. Notice when n is odd it is the number of rose petals. When n is even, there are 2n rose petals.
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Symmetry
• Sometimes symmetry can help with graphing equations in polar coordinates. This leads to the following theorem.
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Example 8
• Sketch the graph of r = a(1 – cos) in polar coordinates assuming a to be a positive constant.
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Example 8
• Sketch the graph of r = a(1 – cos) in polar coordinates assuming a to be a positive constant.
• Observe that replacing with – does not alter the equation, so we know that the graph will be symmetric to the polar axis. Thus, if we graph the upper half, we have the lower half.
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Example 8
• Sketch the graph of r = a(1 – cos) in polar coordinates assuming a to be a positive constant.
• We will now plot points. This graph is called a cardioid (from the Greek word meaning heart).
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Families of Curves
• Equations with any of the four forms below represent polar curves called limacons (from the Latin word “limax” for a snail-like creature that is commonly called a slug). There are four possible shapes for a limacon that are determined by the ratio a/b. If a = b that is a cardioid.
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Graphing
• Look at the following relationships that will always hold. This will make it easy to graph these figures. Notice how the cosine graph is along the x-axis and the sine is along the y-axis.
r = 1 + 2cos r = 1 – sin r = 2 – cos
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Families of Circles
• There are three types of circles. Again, memorize this to make graphing easy.
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Families of Spirals
• A spiral is a curve that coils around a central point. Spirals generally have “left-handed” and “right-handed” versions that coil in opposite directions. Below are some common types of spirals.
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Lemniscates
• The graph of a lemniscate (from the Greek word lemniscos for a looped ribbon resembling the number 8) is pictured below. If it was in terms of sine, it would along the y-axis.
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Homework
• Pages 728-729• 1-11 odd• 17,19• 21-42 multiples of 3