HW # 46 126 97 , 281 63,64 , 355 37,38 , 410 17 Row 3 Do Now Find a set of parametric equations to represent the graph of y = -2x + 1 using the parameters (a) t = x and (b) t = 2 – x. x = t, y = -2t + 1 x = 2 – t, y = -2x + 1 = -2 (2 - t) + 1= -4 + 2t + 1= 2t -3

Polar Coordinates

Objectives Plot points in the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa.

Introduction

Introduction We 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, we will study a different system called the polar coordinate system.

Introduction To form the polar coordinate system in the plane, fix a point O, called the pole (or origin), and construct from O an initial ray called the polar axis, as shown below. Then each point P in the plane can be assigned polar coordinates (r,  ) as follows. 1. r = directed distance from O to P 2.  = directed angle, counterclockwise from polar axis to segment

Example 1(a) – Plotting Points in the Polar Coordinate System The point (r,  ) = (2,  /3) lies two units from the pole on the terminal side of the angle  =  /3, as shown in Figure 10.45. Figure 10.45

Example 1(b) – Plotting Points in the Polar Coordinate System cont’d The point (r,  ) = (3, – /6) lies three units from the pole on the terminal side of the angle  = – /6, as shown in Figure 10.46. Figure 10.46

Example 1(c) – Plotting Points in the Polar Coordinate System cont’d The point (r,  ) = (3, 11 /6) coincides with the point (3, – /6), as shown in Figure 10.47. Figure 10.47

Introduction In rectangular coordinates, each point (x, y) has a unique representation. This is not true for polar coordinates. For instance, the coordinates (r,  ) and (r,  + 2) represent the same point, as illustrated in Example 1. Another way to obtain multiple representations of a point is to use negative values for r. Because r is a directed distance, the coordinates (r,  ) and (–r,  + ) represent the same point.

Introduction In general, the point (r,  ) can be represented as (r,  ) = (r,   2n) or (r,  ) = (–r,   (2n + 1)) where n is any integer. Moreover, the pole is represented by (0,  ) where  is any angle.

Coordinate Conversion

Coordinate Conversion To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the origin, as shown in Figure 10.48. Because (x, y) lies on a circle of radius r, it follows that r 2 = x2 + y2. Figure 10.48

Coordinate Conversion Moreover, for r > 0, the definitions of the trigonometric functions imply that You can show that the same relationships hold for r < 0.

Example 3 – Polar-to-Rectangular Conversion Convert to rectangular coordinates. Solution: For the point (r,  ) = , you have the following.

Example 3 – Solution cont’d The rectangular coordinates are (x, y) = (See Figure 10.49.) Figure 10.49

Equation Conversion

Equation Conversion To convert a rectangular equation to polar form, replace x by r cos  and y by r sin . For instance, the rectangular equation y = x2 can be written in polar form as follows. y = x2 r sin  = (r cos  )2 r = sec  tan  Rectangular equation Polar equation Simplest form

Equation Conversion Converting a polar equation to rectangular form requires considerable ingenuity. Example 5 demonstrates several polar-to-rectangular conversions that enable you to sketch the graphs of some polar equations.

Example 5 – Converting Polar Equations to Rectangular Form Convert each polar equation to a rectangular equation. a. r = 2 b. c. r = sec 

Example 5(a) – Solution The graph of the polar equation r = 2 consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of 2, as shown in Figure 10.51. You can confirm this by converting to rectangular form, using the relationship r 2 = x2 + y2. r = 2 r 2 = 22 x2 + y2 = 2 2 Figure 10.51 Polar equation Rectangular equation

Example 5(b) – Solution The graph of the polar equation cont’d The graph of the polar equation =  /3 consists of all points on the line that makes an angle of  /3 with the positive polar axis, as shown in Figure 10.52. To convert to rectangular form, make use of the relationship tan  = y/x. Figure 10.52 Rectangular equation Polar equation

Example 5(c) – Solution cont’d The graph of the polar equation r = sec  is not evident by simple inspection, so convert to rectangular form by using the relationship r cos  = x. r = sec  r cos  = 1 x = 1 Now you see that the graph is a vertical line, as shown in Figure 10.53. Polar equation Rectangular equation Figure 10.53