600 CHAPTER 9 Polar Coordinates; Vectors Figure 1 x O Pole Polar axis y Fern Hunt (1948-present) Credit: Courtesy of The History Makers Fern Hunt is a mathematician whose research interests are focused on applied probability and dynamical systems, which are mathematical models that describe different types of movement. In addition to mathematical modeling, Hunt has also performed research in biomathematics to study genetic variation and patterns in bacteria. 9.1 Polar Coordinates Now Work the ‘Are You Prepared?’ problems on page 607. • Rectangular Coordinates (Section 1.1, p. 2) • Definition of the Trigonometric Functions (Section 6.2, pp. 397–407) • The Distance Formula (Section 1.2, pp. 13–16) • Inverse Tangent Function (Section 7.1, pp. 477–479) • Completing the Square (Section A.3, p. A29) • Angles; Degree Measure; Radian Measure (Section 6.1, pp. 383–386) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Plot Points Using Polar Coordinates (p. 600) 2 Convert from Polar Coordinates to Rectangular Coordinates (p. 602) 3 Convert from Rectangular Coordinates to Polar Coordinates (p. 604) 4 Transform Equations between Polar and Rectangular Forms (p. 606) So far, we have always used a system of rectangular coordinates to plot points in the plane. However, there are times when rectangular coordinates are too cumbersome and inefficient, and the need arises to describe a relationship between two variables using a different coordinate system, called polar coordinates. In many instances, polar coordinates offer certain advantages over rectangular coordinates. In a rectangular coordinate system, you will recall, a point in the plane is represented by an ordered pair of numbers ( ) x y , , where x and y equal the signed distances of the point from the y -axis and the x -axis, respectively. In a polar coordinate system, we select a point, called the pole , and then a ray with vertex at the pole, called the polar axis . See Figure 1. Comparing the rectangular and polar coordinate systems, note that the origin in rectangular coordinates coincides with the pole in polar coordinates, and the positive x -axis in rectangular coordinates coincides with the polar axis in polar coordinates. 1 Plot Points Using Polar Coordinates A point P in a polar coordinate system is represented by an ordered pair θ ( ) r, of numbers. If > r 0, then r is the distance of the point from the pole; θ is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.We call the ordered pair θ ( ) r, the polar coordinates of the point. See Figure 2. As an example, suppose that a point P has polar coordinates π ( ) 2, 4 . Locate P by first drawing an angle of π 4 radians, placing its vertex at the pole and its initial side along the polar axis. Then go out a distance of 2 units along the terminal side of the angle to reach the point P . See Figure 3. Figure 2 O Pole Polar axis P 5 (r, u) r u Figure 3 P 5 2, ( ) p– 4 O Pole Polar axis 2 p– 4 In using polar coordinates θ ( ) r, , it is possible for r to be negative. When this happens, instead of the point being on the terminal side of θ, it is on the ray from the pole extending in the direction opposite the terminal side of θ at a distance r units from the pole. See Figure 4 on the next page for an illustration. For example, to plot the point π ( ) −3, 2 3 , use the ray in the opposite direction of π2 3 and go out − = 3 3 units along that ray. See Figure 5 on the next page.
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