1. Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.

2. Polar Coordinates and Polar Graphs 2015 Copyright © Cengage Learning. All rights reserved. 10.4

3. Understand the polar coordinate system. Rewrite rectangular coordinates and equations in polar form and vice versa. Sketch the graph of an equation given in polar form. Find the slope of a tangent line to a polar graph. Identify several types of special polar graphs. Objectives

4. 10.4 Polar Coordinates Greg Kelly, Hanford High School, Richland, Washington

5. Polar Coordinates

6. You may represent graphs as collections of points (x, y) on the rectangular coordinate system. The corresponding equations for these graphs have been in either rectangular or parametric form. In this section you will study a coordinate system called the polar coordinate system. Polar Coordinates

7. One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. Polar Coordinates

8. 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 10.36. Figure 10.36 Polar Coordinates

9. Then each point P in the plane can be assigned polar coordinates (r, θ), as follows. r = directed distance from O to P θ = directed angle, counterclockwise from polar axis to segment Figure 10.37 shows three points on the polar coordinate system. Figure 10.37

10. (Circle centered at the origin) (Line through the origin) Some curves are easier to describe with polar coordinates:

11. 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 10.38. 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 If r < 0, you can show that the same relationships hold. Figure 10.38 Coordinate Conversion

13. More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are: Polar Coordinates are not unique.

14. a. Convert the point (2, π) to rectangular form. The rectangular coordinates are (x, y) = (–2, 0). b. Convert the point to rectangular form. Polar-to-Rectangular Conversion The rectangular coordinates are (x, y) =

15. a. Convert the point (1,1) to polar form. Rectangular to-Polar Conversion b. Convert the point (-3,4) to polar form.

16. a. Convert the equation to polar form. Equation Conversion b. Convert the equation to rectangular form.

17. Polar Graphs

18. Tests for Symmetry: x-axis: If (r,  ) is on the graph,so is (r, -  ).

19. Tests for Symmetry: y-axis: If (r,  ) is on the graph,so is (r,  -  )or (-r, -  ).

20. Tests for Symmetry: origin: If (r,  ) is on the graph,so is (-r,  )or (r,  +  ).

21. Tests for Symmetry: If a graph has two symmetries, then it has all three: 

22. Describe the graph of each polar equation. Confirm each description by converting to a rectangular equation. Example 3 – Graphing Polar Equations

23. 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. [See Figure 10.41(a).] You can confirm this by using the relationship r 2 = x 2 + y 2 to obtain the rectangular equation Figure 10.41(a) Example 3(a) – Solution

24. The graph of the polar equation θ = π/3 consists of all points on the line that makes an angle of π/3 with the positive x-axis. [See Figure 10.41(b).] You can confirm this by using the relationship tan θ = y/x to obtain the rectangular equation Figure 10.41(b) Example 3(a) – Solution

25. The graph of the polar equation r = sec θ is not evident by simple inspection, so you can begin by converting to rectangular form using the relationship r cos θ = x. From the rectangular equation, you can see that the graph is a vertical line. Figure 10.41(c) Example 3(c) – Solution

26. The graph of shown in Figure 10.42 was produced with a graphing calculator in parametric mode. This equation was graphed using the parametric equations with the values of θ varying from –4π to 4π. This curve is of the form r = aθ and is called a spiral of Archimedes. Figure 10.42 Polar Graphs

27. Slope and Tangent Lines Not tested, optional

28. To find the slope of a polar curve: 1. Write the equation in parametric form using: x = r cos θ and y = r sin θ 2. Differentiate x and y with respect to θ. We use the product rule here.

29. To find the slope of a polar curve: Product rule

30. Example:

31. To find the slope of a tangent line to a polar graph, consider a differentiable function given by r = f(θ). To find the slope in polar form, use the parametric equations x = r cos θ = f(θ) cos θ and y = r sin θ = f(θ) sin θ. Using the parametric form of dy/dx given in Theorem 10.7, you have Slope and Tangent Lines

32. Figure 10.45 Slope and Tangent Lines

33. From Theorem 10.11, you can make the following observations. 1. Solutions to yield horizontal tangents, provided that 2. Solutions to yield vertical tangents, provided that If dy/dθ and dx/dθ are simultaneously 0, no conclusion can be drawn about tangent lines. Slope and Tangent Lines

34. Find the horizontal and vertical tangent lines of r = sin θ, 0 ≤ θ ≤ π. Solution: Begin by writing the equation in parametric form. x = r cos θ = sin θ cos θ and y = r sin θ = sin θ sin θ = sin 2 θ Next, differentiate x and y with respect to θ and set each derivative equal to 0. Example 5 – Finding Horizontal and Vertical Tangent Lines

35. So, the graph has vertical tangent lines at and and it has horizontal tangent lines at (0, 0) and (1, π/2). Figure 10.46 Example 5 – Solution to finding horizontal and vertical tangent lines to r = sin θ.

36. Ex: Find any lines tangent to the pole for Solution:

38. Special Polar Graphs

39. Several important types of graphs have equations that are simpler in polar form than in rectangular form. For example, the polar equation of a circle having a radius of a and centered at the origin is simply r = a. Several other types of graphs that have simpler equations in polar form are shown below. Special Polar Graphs

41. Homework Section 10.4 Day 1: pg. 736, #1-51 odd Day 2: MMM pgs 162-163