10.5. Polar Coordinates. 714 Chapter 10: Conic Sections and Polar Coordinates. Definition of Polar Coordinates

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1 71 Chapter 1: Conic Sections and Polar Coordinates 1.5 Polar Coordinates rigin (pole) r P(r, ) Initial ra FIGURE 1.5 To define polar coordinates for the plane, we start with an origin, called the pole, and an initial ra. In this section, we stud polar coordinates and their relation to Cartesian coordinates. While a point in the plane has just one pair of Cartesian coordinates, it has infinitel man pairs of polar coordinates. This has interesting consequences for graphing, as we will see in the net section. Definition of Polar Coordinates To define polar coordinates, we first fi an origin (called the pole) and an initial ra from (Figure 1.5). Then each point P can be located b assigning to it a polar coordinate pair sr, ud in which r gives the directed distance from to P and u gives the directed angle from the initial ra to ra P. Polar Coordinates Psr, ud Directed distance from to P Directed angle from initial ra to P 11 FIGURE 1. unique. P 2, P 2, 11 Initial ra Polar coordinates are not As in trigonometr, u is positive when measured counterclockwise and negative when measured clockwise. The angle associated with a given point is not unique. For instance, the point 2 units from the origin along the ra u = p> has polar coordinates r = 2, u = p>. It also has coordinates r = 2, u = -11p> (Figure 1.). There are occasions when we wish to allow r to be negative. That is wh we use directed distance in defining Psr, ud. The point Ps2, 7p>d can be reached b turning 7p> radians counterclockwise from the initial ra and going forward 2 units (Figure 1.7). It can also be reached b turning p> radians counterclockwise from the initial ra and going backward 2 units. So the point also has polar coordinates r = -2, u = p>. 7 7 P 2, 7 P 2, FIGURE 1.7 r-values. Polar coordinates can have negative

2 1.5 Polar Coordinates 715 EXAMPLE 1 Finding Polar Coordinates Find all the polar coordinates of the point Ps2, p>d. Solution We sketch the initial ra of the coordinate sstem, draw the ra from the origin that makes an angle of p> radians with the initial ra, and mark the point s2, p>d (Figure 1.8). We then find the angles for the other coordinate pairs of P in which r = 2 and r = Initial ra 2, 2, 5 2, 7 etc. 5 FIGURE 1.8 The point Ps2, p>d has infinitel man polar coordinate pairs (Eample 1). For r = 2, the complete list of angles is For r = -2, the angles are - 5p, - 5p ; 2p, - 5p ; p, - 5p ; p, Á. The corresponding coordinate pairs of P are p, p ; 2p, p ; p, p ; p, Á. a2, p + 2npb, n =, ;1, ;2, Á and a-2, - 5p + 2npb, n =, ;1, ;2, Á. r a a FIGURE 1.9 circle is r = a. The polar equation for a When n =, the formulas give s2, p>d and s -2, -5p>d. When n = 1, the give s2, 1p>d and s -2, 7p>d, and so on. Polar Equations and Graphs If we hold r fied at a constant value r = a Z, the point Psr, ud will lie ƒ a ƒ units from the origin. As u varies over an interval of length 2p, P then traces a circle of radius ƒ a ƒ centered at (Figure 1.9). If we hold u fied at a constant value u = u and let r var between - q and q, the point Psr, ud traces the line through that makes an angle of measure u with the initial ra.

3 71 Chapter 1: Conic Sections and Polar Coordinates 1 r 2, 2 Equation Graph 1 2 r = a u = u Circle radius ƒ a ƒ centered at Line through making an angle u with the initial ra 2, r 2 EXAMPLE 2 Finding Polar Equations for Graphs r = 1 and r = -1 are equations for the circle of radius 1 centered at. u = p>, u = 7p>, and u = -5p> are equations for the line in Figure 1.8. Equations of the form r = a and u = u can be combined to define regions, segments, and ras. EXAMPLE Identifing Graphs Graph the sets of points whose polar coordinates satisf the following conditions., r 1 r 2 and u p 2 - r 2 and u = p (d) (d) r and u = p 2p u 5p sno restriction on rd Solution The graphs are shown in Figure 1.. FIGURE 1. The graphs of tpical inequalities in r and u (Eample ). Relating Polar and Cartesian Coordinates When we use both polar and Cartesian coordinates in a plane, we place the two origins together and take the initial polar ra as the positive -ais. The ra u = p>2, r 7, becomes the positive -ais (Figure 1.1). The two coordinate sstems are then related b the following equations. Ra 2 P(, ) P(r, ) Common r origin, r Initial ra FIGURE 1.1 The usual wa to relate polar and Cartesian coordinates. Equations Relating Polar and Cartesian Coordinates = r cos u, = r sin u, = r 2 The first two of these equations uniquel determine the Cartesian coordinates and given the polar coordinates r and u. n the other hand, if and are given, the third equation gives two possible choices for r (a positive and a negative value). For each selection, there is a unique u H [, 2pd satisfing the first two equations, each then giving a polar coordinate representation of the Cartesian point (, ). The other polar coordinate representations for the point can be determined from these two, as in Eample 1.

4 1.5 Polar Coordinates 717 EXAMPLE Equivalent Equations Polar equation r cos u = 2 r 2 cos u sin u = r 2 cos 2 u - r 2 sin 2 u = 1 r = 1 + 2r cos u r = 1 - cos u Cartesian equivalent = 2 = 2-2 = = = With some curves, we are better off with polar coordinates; with others, we aren t. (, ) 2 ( ) 2 9 or r sin FIGURE 1.2 The circle in Eample 5. EXAMPLE 5 Converting Cartesian to Polar Find a polar equation for the circle 2 + s - d 2 = 9 (Figure 1.2). Solution = 9 Epand s - d The 9 s cancel. - = r 2 - r sin u = = r 2 r = or r - sin u = r = sin u Includes both possibilities We will sa more about polar equations of conic sections in Section 1.8. EXAMPLE Converting Polar to Cartesian Replace the following polar equations b equivalent Cartesian equations, and identif their graphs. r cos u = - r 2 = r cos u r = 2 cos u - sin u Solution We use the substitutions r cos u =, r sin u =, r 2 = r cos u = - The Cartesian equation: r cos u = - = - The graph: Vertical line through = - on the -ais r 2 = r cos u The Cartesian equation: r 2 = r cos u = = = s - 2d = The graph: Circle, radius 2, center sh, kd = s2, d Completing the square

5 718 Chapter 1: Conic Sections and Polar Coordinates r = 2 cos u - sin u The Cartesian equation: rs2 cos u - sin ud = 2r cos u - r sin u = 2 - = = 2 - The graph: Line, slope m = 2, -intercept b = -