Chap10_Sec3 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

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PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES 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 two perpendicular axes. PARAMETRIC EQUATIONS & POLAR COORDINATES
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Here, we describe a coordinate system introduced by Newton, called the polar coordinate system. It is more convenient for many purposes. PARAMETRIC EQUATIONS & POLAR COORDINATES
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10.3 Polar 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 right corresponding to the positive x -axis in Cartesian coordinates.
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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 counterclockwise direction 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 same quadrant 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, 5 π /4) b. (2, 3 π ) c. (2, –2 π /3) d. (–3, 3 π /4) Example 1
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POLAR COORDINATES The point (1, 5 π /4) is plotted here. Example 1 a
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The point (2, 3 π ) is plotted. Example 1 b POLAR COORDINATES
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POLAR COORDINATES The point (2, –2 π /3) is plotted. Example 1 c
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POLAR COORDINATES The point (–3, 3 π /4) is plotted. It is is located three units from the pole in the fourth quadrant. This is because the angle 3 π /4 is in the second quadrant and r = -3 is negative. Example 1 d
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CARTESIAN VS. POLAR COORDINATES In the Cartesian coordinate system, every point 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, 5 π /4) in Example 1 a could be written as: (1, –3 π /4), (1, 13 π /4), or (–1, π /4).
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CARTESIAN & POLAR COORDINATES In fact, as a complete counterclockwise rotation is given by an angle 2 π , the point
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