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10 Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved. 10.4 Polar Coordinates and Polar Graphs 2015 Copyright © Cengage Learning. All rights reserved. Objectives 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. 10.4 Polar Coordinates Greg Kelly, Hanford High School, Richland, Washington Polar Coordinates Polar Coordinates 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 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. r Initial ray A polar coordinate pair r , determines the location of a point. Polar Coordinates 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 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 OP Figure 10.37 shows three points on the polar coordinate system. Figure 10.37 Some curves are easier to describe with polar coordinates: r a (Circle centered at the origin) o (Line through the origin) 1 r 2 0 2 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 10.38. Because (x, y) lies on a circle of radius r, it follows that r2 = x2 + y2. Moreover, for r > 0 the definitions of the trigonometric functions imply that Figure 10.38 and If r < 0, you can show that the same relationships hold. Coordinate Conversion Polar Coordinates are not unique. More than one coordinate pair can refer to the same point. 2,30 o 2 210 o 30 2, 210 o 150o o 2, 150 o All of the polar coordinates of this point are: o o 2,30 n 360 2, 150 o n 360 o n 0, 1, 2 ... Polar-to-Rectangular Conversion a. Convert the point (2, π) to rectangular form. The rectangular coordinates are (x, y) = (–2, 0). b. Convert the point 3, to rectangular form. 6 The rectangular coordinates are (x, y) = Rectangular to-Polar Conversion a. Convert the point (1,1) to polar form. One set of polar coordinates is (r, ) Another set is (r, ) 2, 5 . 4 b. Convert the point (-3,4) to polar form. 5, 2.214 2, . 4 Equation Conversion a. Convert the equation to polar form. tan r cos b. Convert the equation to rectangular form. Polar Graphs Tests for Symmetry: x-axis: If (r, ) is on the graph, so is (r, -). 1 r r 2cos 0 1 2 r -1 Tests for Symmetry: y-axis: If (r, ) is on the graph, so is (r, -) or (-r, -). 2 r r 2sin 1 -1 r 0 1 Tests for Symmetry: origin: If (r, ) is on the graph, so is (-r, ) or (r, +) . 2 tan r cos 1 r -2 0 -1 r 1 2 -1 -2 Tests for Symmetry: If a graph has two symmetries, then it has all three: 2 1 r 2cos 2 -2 -1 0 1 2 -1 -2 Example 3 – Graphing Polar Equations Describe the graph of each polar equation. Confirm each description by converting to a rectangular equation. Example 3(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. [See Figure 10.41(a).] You can confirm this by using the relationship r2 = x2 + y2 to obtain the rectangular equation Figure 10.41(a) Example 3(a) – Solution 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(c) – Solution 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) Polar Graphs 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 Slope and Tangent Lines Not tested, optional 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. dy dy d dx dx d d r sin d d r cos d r sin r cos r cos r sin We use the product rule here. To find the slope of a polar curve: Product rule dy dy d dx dx d d r sin d d r cos d r sin r cos r cos r sin dy r sin r cos dx r cos r sin Example: r sin r 1 cos dy r sin r cos dx r cos r sin sin sin 1 cos cos Slope sin cos 1 cos sin sin cos cos sin cos sin sin cos 2 2 sin cos cos 2sin cos sin 2 2 cos 2 cos sin 2 sin Slope and Tangent Lines Figure 10.45 Slope and Tangent Lines 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. Example 5 – Finding Horizontal and Vertical Tangent Lines 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 = sin2 θ Next, differentiate x and y with respect to θ and set each derivative equal to 0. Example 5 – Solution to finding horizontal and vertical tangent lines to r = sin θ. 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 Ex: Find any lines tangent to the pole for f 2cos3 Solution: 5 Since f 2 cos 3 0 when , , and , 6 2 6 and the derivative f 6sin 3 0 for any of these values, 5 f has three tangent lines at , , and . 6 2 6 Special Polar Graphs Special Polar Graphs 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 Homework • Section 10.4 Day 1: pg. 736, #1-51 odd • Day 2: MMM pgs 162-163