Copyright © Cengage Learning. All rights reserved. 9 Topics in Analytic Geometry

Copyright © Cengage Learning. All rights reserved. 9.5 Polar Coordinates

3 What You Should Learn Plot points and find multiple representations of 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

4 Introduction

5 Representation of graphs of equations as collections of points (x, y), where x and y represent the directed distances from the coordinate axes to the point (x, y) is called rectangular coordinate system. In this section, you will study a second coordinate system called the polar coordinate system.

6 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 in Figure Figure 9.59

7 Introduction Then each point in the plane can be assigned polar coordinates (r,  ) as follows. 1. r = directed distance from O to P 2.  = directed angle, counterclockwise from the polar axis to segment OP

8 Example 1 – Plotting Points in the Polar Coordinate System a. The point lies two units from the pole on the terminal side of the angle as shown in Figure Figure 9.60

9 Example 1 – Plotting Points in the Polar Coordinate System b. The point lies three units from the pole on the terminal side of the angle as shown in Figure Figure 9.61

10 Example 1 – Plotting Points in the Polar Coordinate System c. The point coincides with the point, as shown in Figure Figure 9.62

11 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.

12 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.

13 Example 2 – Multiple Representations of Points Plot the point and find three additional polar representations of this point, using – 2  <  < 2 .

14 Example 2 – Solution The point is shown in Figure Figure 9.63

15 Example 2 – Solution Three other representations are as follows. Add 2  to  Replace r by –r, subtract  from  Replace r by –r add  to  cont’d

16 Coordinate Conversion

17 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 Figure 9.64

18 Coordinate Conversion Because (x, y) lies on a circle of radius r, it follows that r 2 = x 2 + y 2. Moreover, for r > 0, the definitions of the trigonometric functions imply that and You can show that the same relationships hold for r < 0.

19 Example 3 – Polar-to-Rectangular Conversion Convert the point (2,  ) to rectangular coordinates. Solution: For the point (r,  ) = (2,  ), you have the following. x = r cos  = 2 cos  = –2 x = r sin  = 2 sin  = 0 The rectangular coordinates are (x, y) = (–2, 0). (See Figure 9.65.) Figure 9.65

20 Equation Conversion

21 Equation Conversion Point conversion from the polar to the rectangular system is straightforward, whereas point conversion from the rectangular to the polar system is more involved. For equations, the opposite is true. To convert a rectangular equation to polar form, you simply replace x by r cos  and y by r sin . For instance, the rectangular equation y = x 2 can be written in polar form as follows. y = x 2 Rectangular equation

22 Equation Conversion r sin  = (r cos  ) 2 r = sec  tan  On the other hand, converting a polar equation to rectangular form requires considerable ingenuity. Polar equation Simplest form

23 Example 5 – Converting Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r = 2b.c. r = sec  Solution: a. 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 Figure 9.67

24 Example 5 – Solution You can confirm this by converting to rectangular form, using the relationship r 2 = x 2 + y 2. b. The graph of the polar equation consists of all points on the line that makes an angle of  3 with the positive x-axis, as shown in Figure cont’d r 2 = 2 2 Figure 9.68

25 Example 5 – Solution To convert to rectangular form, you make use of the relationship tan  = y  x. c. The graph of the polar equation r = sec  is not evident by simple inspection, so you convert to rectangular form by using the relationship r cos  = x. cont’d

26 Example 5 – Solution Now you can see that the graph is a vertical line, as shown in Figure cont’d Figure 9.69