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Polar Coordinates and Graphs of Polar Equations

                                    Name: ___________________
                                    Period: __________________
                                    Date: ____________________
Across
To form the polar coordinate system in the plane, fix a point O, called the ____ (or origin), and construct from ) an initial ray called the polar axis.
The graph in Figure 10.57, like the one in Figure 10.55, is a(n) ______.
For r > 0, the definitions of the trigonometric functions imply that ___ θ = y / x
Because (x, y) lies on a circle of ______ r, it follows that r^2 = x^2 + y^2.
Two additional aids to sketching graphs of polar equations involve knowing the θ-values for which |r| is _______ and knowing the θ-values for which r = 0.
Some curves reach their _____ and maximum r-values at more than one point, as shown in Example 4.
You can ________ the graph in Figure 10.54 by converting the polar equation to rectangular form and then sketching the graph of the rectangular equation.
Had you known about this symmetry and retracing ahead of time, you could have used ______ points.
The _____ of r = g(cos θ) is symmetric with respect to the polar axis.
Each point P in the plane can be assigned polar ___________ (r, θ) as follows: 1. r = directed distance from O to P; 2. θ = directed angle, counterclockwise from polar axis to segment OP.
In rectangular coordinates, each point (x, y) has a(n) ______ representation. This is not true for polar coordinates.
By plotting these points and using specified symmetry, zeros, and maximum values, you can obtain the graph, as shown below. This graph is called a(n) ____ curve, and each loop on the graph is called a petal.
Another way to obtain multiple representations of a point is to use ________ values for r.
For r > 0, the definitions of the trigonometric functions imply that ___ θ = x / r
A type of graph with polar equations in the form r^2 = a^2 sin 2θ or r^2 = a^2 cos 2θ
To _______ a rectangular equation to polar form, replace x by r cos θ and y by r sin θ.
To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the ________ x-axis and the pole with the origin.
In Figure 10.54 on the preceding page, note that as θ __________ from 0 to 2π the graph is traced out twice.
The coordinates (r, θ) and (r, θ + 2π) represent the ____ point, as illustrated in Example 1.
Down
To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the ______.
So far, you have been representing graphs of equations as collections of points (x, y) in the rectangular coordinate system, where x and y represent the directed distances from the coordinate axes to the point (x, y). In this section, you will study a different system called the _____ coordinate system.
Each point P in the plane can be assigned polar coordinates (r, θ) as follows: 1. r = directed distance from O to P; 2. θ = directed _____, counterclockwise from polar axis to segment OP.
Recall from Section 4.2 that the cosine function is even and the sine function is ___. That is, sin(-θ) = -sin(θ).
To form the polar coordinate system in the plane, fix a point O, called the pole (or origin), and construct from ) an initial ray called the polar ____.
Note that the graph is _________ with respect to the line θ = π/2.
The three tests for symmetry in polar coordinates listed on page 748 are sufficient to _________ symmetry, but they are not necessary.
You can use polar coordinates in mathematical modeling. For instance, in Exercise 127 on page 746, you will use polar coordinates to model the path of a passenger car on a ______ wheel.
In previous chapters, you learned how to sketch graphs in rectangular coordinate systems. You began with the basic point-________ method. Then you used sketching aids such as symmetry, intercepts, asymptotes, periods, and shifts to further investigate the natures of graphs. This section approaches curve sketching in the polar coordinate system similarly, beginning with a demonstration of point _______.
Each point P in the plane can be assigned polar coordinates (r, θ) as follows: 1. r = directed _________ from O to P; 2. θ = directed angle, counterclockwise from polar axis to segment OP.
Note how the negative r-values determine the _____ loop of the graph in Figure 10.57.
You can use graphs of polar equations in mathematical modeling. For instance, in Exercise 69 on page 754, you will graph the pickup pattern of a(n) ____________ in a polar coordinate system.
The graph of r = f(sin θ) is symmetric with respect to the line θ = π/2.
Converting a polar equation to rectangular form requires considerable _________.
Because r is a(n) ________ distance, the coordinates (r, θ) and (-r, θ + π) represent the same point.
There are _____ important types of symmetry to consider in polar curve sketching: Symmetry with Respect to the Line θ = π/2, Symmetry with Respect to the Polar Axis, and Symmetry with Respect to the Pole
By plotting these points and using specified symmetry, zeros, and maximum values, you can obtain the graph, as shown below. This graph is called a rose curve, and each loop on the graph is called a(n) _____.
For r > 0, the definitions of the trigonometric functions imply that ___ θ = y / r
Note in Example 2 that cos(-θ) = cos θ. This is because the cosine function is _____.