1 _7.qxd /8/5 9: AM Page 779 Section.7 Polar Coordinates Polar Coordinates What ou should learn Plot points on the polar coordinate sstem. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and vice versa. Wh ou should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Exercises 8 on page 78, ou are asked to find multiple representations of polar coordinates. Introduction So far, ou have been representing graphs of equations as collections of points x, on the rectangular coordinate sstem, where x and represent the directed distances from the coordinate axes to the point x,. In this section, ou will stud a different sstem called the polar coordinate sstem. To form the polar coordinate sstem in the plane, fix a point O, called the pole (or origin), and construct from O an initial ra called the polar axis, as shown in Figure.57. Then each point P in the plane can be assigned polar coordinates r, as follows.. r directed distance from O to P. directed angle, counterclockwise from polar axis to segment OP O r = directed distance θ FIGURE.57 P= ( r, θ) = directed angle Polar axis Example Plotting Points on the Polar Coordinate Sstem a. The point r,, lies two units from the pole on the terminal side of the angle as shown in Figure.58. b. The point r,, lies three units from the pole on the terminal side of the angle as shown in Figure.59. c. The point r,, coincides with the point,, as shown in Figure..,, θ =, FIGURE.58 θ = FIGURE.59, Now tr Exercise. FIGURE., θ =
2 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr, θ =, = (, 5 ) =, 7 =, =... FIGURE. Pole Exploration Most graphing calculators have a polar graphing mode. If ours does, graph the equation r. (Use a setting in which x and.) You should obtain a circle of radius. a. Use the trace feature to cursor around the circle. Can ou locate the point, 5? b. Can ou find other polar representations of the point, 5? If so, explain how ou did it. (Origin) FIGURE. θ r x (r, θ) (x, ) x Polar axis (x-axis) In rectangular coordinates, each point x, has a unique representation. This is not true for polar coordinates. For instance, the coordinates r, and r, represent the same point, as illustrated in Example. Another wa 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. In general, the point r, can be represented as r, r, ± n or r, r, ± n where n is an integer. Moreover, the pole is represented b,, where is an angle. Example Multiple Representations of Points Plot the point, and find three additional polar representations of this point, using < <. The point is shown in Figure.. Three other representations are as follows., 5,, 7,,, Now tr Exercise. Coordinate Conversion Add to. Replace r b r; subtract Replace r b r; add to. from. 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.. Because x, lies on a circle of radius r, it follows that r x. Moreover, for r >, the definitions of the trigonometric functions impl that tan x, cos x r, and sin r. If r <, ou can show that the same relationships hold. Coordinate Conversion The polar coordinates r, are related to the rectangular coordinates x, as follows. Polar-to-Rectangular Rectangular-to-Polar x r cos r sin tan x r x
3 _7.qxd /8/5 9: AM Page 78 Example Section.7 Polar Coordinates 78 Polar-to-Rectangular Conversion (, r θ) = (, ) (, x ) = (, ) FIGURE. Activities. Find three additional polar representations of the point (, r θ) =, (, x ) =,,., Answer:,, and, 5,. Convert the point, from rectangular to polar form. Answer: 5,. or 5,.779 x Convert each point to rectangular coordinates. a., b., a. For the point r,,, ou have the following. x r cos cos r sin sin The rectangular coordinates are x,,. (See Figure..) b. For the point r,, ou have the following. x cos sin, The rectangular coordinates are Now tr Exercise. x,,. Example Rectangular-to-Polar Conversion (x, ) = (, ) (r, θ) =, FIGURE. (x, ) = (, ) FIGURE.5 (r, θ) =, Convert each point to polar coordinates. a., b., a. For the second-quadrant point x,,, ou have tan x Because lies in the same quadrant as x,, use positive r. r x So, one set of polar coordinates is r,,, as shown in Figure.. b. Because the point x,, lies on the positive -axis, choose. and r. This implies that one set of polar coordinates is r,,, as shown in Figure.5. Now tr Exercise 9.
4 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr Multiple representations of points and equations in the polar sstem can cause confusion. You ma want to discuss several examples. Activities. Convert the polar equation r cos to rectangular form. Answer: x x. Convert the rectangular equation x to polar form. Answer: r sec Equation Conversion B comparing Examples and, ou can see that point conversion from the polar to the rectangular sstem is straightforward, whereas point conversion from the rectangular to the polar sstem is more involved. For equations, the opposite is true. To convert a rectangular equation to polar form, ou simpl replace x b r cos and b r sin. For instance, the rectangular equation x can be written in polar form as follows. x r sin r cos r sec tan Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuit. Example 5 demonstrates several polar-to-rectangular conversions that enable ou to sketch the graphs of some polar equations. Example 5 Converting Polar Equations to Rectangular Form FIGURE. Describe the graph of each polar equation and find the corresponding rectangular equation. a. r b. c. r sec a. The graph of the polar equation r 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, as shown in Figure.. You can confirm this b converting to rectangular form, using the relationship r x. r r x FIGURE.7 b. The graph of the polar equation consists of all points on the line that makes an angle of with the positive polar axis, as shown in Figure.7. To convert to rectangular form, make use of the relationship tan x. tan x c. The graph of the polar equation r sec is not evident b simple inspection, so convert to rectangular form b using the relationship r cos x. r sec r cos x FIGURE.8 Now ou see that the graph is a vertical line, as shown in Figure.8. Now tr Exercise 5.
5 _7.qxd /8/5 :9 AM Page 78 Section.7 Polar Coordinates 78.7 Exercises VOCABULARY CHECK: Fill in the blanks.. The origin of the polar coordinate sstem is called the.. For the point r,, r is the from O to P and is the counterclockwise from the polar axis to the line segment OP.. To plot the point r,, use the coordinate sstem.. The polar coordinates r, are related to the rectangular coordinates x, as follows: x tan r PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at In Exercises 8, plot the point given in polar coordinates and find two additional polar representations of the point, using < <..,.. 5, 7., 5.,..,.57 7., ,. In Exercises 9, a point in polar coordinates is given. Convert the point to rectangular coordinates. 9.,. (, r θ) =, 5.,., (r, θ) =, 5,, (, r θ) =, (r, θ) = (, ) 7.,., 5..5,.. 8.5,.5 In Exercises 7, a point in rectangular coordinates is given. Convert the point to polar coordinates. 7., 8., 9.,., 5.,.,.,., 5., 9. 5, In Exercises 7, use a graphing utilit to find one set of polar coordinates for the point given in rectangular coordinates. 7., 8. 5, 9.,.,,.. 5, 7, In Exercises 8, convert the rectangular equation to polar form. Assume a >.. x 9. x 5.. x 7. x 8. x a 9. x. x 5. x. x. 8x. x 9x 5. x a. x 9a 7. x ax 8. x a
6 _7.qxd /8/5 9: AM Page Chapter Topics in Analtic Geometr In Exercises 9,convert the polar equation to rectangular form. 9. r sin 5. r cos r 5. r 55. r csc 5. r sec 57. r cos 58. r sin 59. r sin. r cos. r. r sin cos. r. r sin cos sin In Exercises 5 7, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. 5. r. r r sec 7. r csc Snthesis 5 True or False? In Exercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. If n for some integer n, then r, and r, represent the same point on the polar coordinate sstem. 7. If r r, then r, and r, represent the same point on the polar coordinate sstem. 7. Convert the polar equation r h cos k sin to rectangular form and verif that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle. 7. Convert the polar equation r cos sin to rectangular form and identif the graph. 75. Think About It (a) Show that the distance between the points r, and r, is r r r r cos. (b) Describe the positions of the points relative to each other for. Simplif the Distance Formula for this case. Is the simplification what ou expected? Explain. (c) Simplif the Distance Formula for 9. Is the simplification what ou expected? Explain. (d) Choose two points on the polar coordinate sstem and find the distance between them. Then choose different polar representations of the same two points and appl the Distance Formula again. Discuss the result. 7. Exploration (a) Set the window format of our graphing utilit on rectangular coordinates and locate the cursor at an position off the coordinate axes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor verticall. (b) Set the window format of our graphing utilit on polar coordinates and locate the cursor at an position off the coordinate axes. Move the cursor horizontall and observe an changes in the displaed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor verticall. (c) Explain wh the results of parts (a) and (b) are not the same. Skills Review In Exercises 77 8, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 77. log x z 78. log x 79. ln xx 8. ln 5x x In Exercises 8 8, condense the expression to the logarithm of a single quantit. 8. log 7 x log 7 8. log 5 a 8 log 5 x 8. ln x lnx 8. ln ln lnx In Exercises 85 9, use Cramer s Rule to solve the sstem of equations x 7 x a b c a b c a b 9c x z x z 5x z 9. x 5 x 5u u 8u x x x 7v v v 5 x x x 9w w w x x In Exercises 9 9, use a determinant to determine whether the points are collinear. 9.,,, 7,, 9.,,,,, 5 9.,,,,.5,.5 9.., 5,.5,,.5, 5 7 5