SECTIN. PLAR CRDINATES 67 _ and so we require that 6n5 be an even multiple of. This will first occur when n 5. Therefore we will graph the entire curve if we specify that. Switching from to t, we have the equations x sin8t5 cos t y sin8t5 sin t t and Figure 8 shows the resulting curve. Notice that this rose has 6 loops. M FIGURE 8 r=sin(8 /5) _ N In Exercise 55 you are asked to prove analytically what we have discovered from the graphs in Figure 9. V EXAMPLE Investigate the family of polar curves given by r c sin. How does the shape change as c changes? (These curves are called limaçons, after a French word for snail, because of the shape of the curves for certain values of c.) SLUTIN Figure 9 shows computer-drawn graphs for various values of c. For c there is a loop that decreases in size as c decreases. When c the loop disappears and the curve becomes the cardioid that we sketched in Example 7. For c between and the cardioid s cusp is smoothed out and becomes a dimple. When c decreases from to, the limaçon is shaped like an oval. This oval becomes more circular as c l, and when c the curve is just the circle r. c=.7 c= c=.7 c=.5 c=. c=.5 c=_ c= c=_. c=_.5 c=_.8 c=_ FIGURE 9 Members of the family of limaçons r=+c sin The remaining parts of Figure 9 show that as c becomes negative, the shapes change in reverse order. In fact, these curves are reflections about the horizontal axis of the corresponding curves with positive c. M. EXERCISES Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r and one with r.. (a), (b), (c),. (a), 7 (b), 6 (c), Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.. (a), (b) (, ) (c),. (a) (s, 5) (b), 5 (c), 76 5 6 The Cartesian coordinates of a point are given. (i) Find polar coordinates r, of the point, where r and. (ii) Find polar coordinates r, of the point, where r and. 5. (a), (b) (, s ) 6. (a) (s, ) (b),
68 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES 7 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 7. r 8. r, 9. r,. r 5,. r,. r,. Find the distance between the points with polar coordinates, and,.. Find a formula for the distance between the points with polar coordinates r, and r,. 5 Identify the curve by finding a Cartesian equation for the curve. 5. r 6. r cos 7. r sin 8. r sin cos 9. r csc. r tan sec 6 Find a polar equation for the curve represented by the given Cartesian equation.. x. x y 9. x y. x y 9 5. x y cx 5 6 5 7 7 8 For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. 7. (a) A line through the origin that makes an angle of 6 with the positive x-axis (b) A vertical line through the point, 6. xy 8. (a) A circle with radius 5 and center, (b) A circle centered at the origin with radius. r 9 sin. r cos 5. r cos 6. r 7. r cos 8. r cos 9 5 The figure shows the graph of r as a function of in Cartesian coordinates. Use it to sketch the corresponding polar curve. 9. r 5. Show that the polar curve r sec (called a conchoid) has the line x as a vertical asymptote by showing that lim r l x. Use this fact to help sketch the conchoid. 5. Show that the curve r csc (also a conchoid) has the line y as a horizontal asymptote by showing that lim r l y. Use this fact to help sketch the conchoid. 5. Show that the curve r sin tan (called a cissoid of Diocles) has the line x as a vertical asymptote. Show also that the curve lies entirely within the vertical strip x. Use these facts to help sketch the cissoid. 5. Sketch the curve x y x y. 55. (a) In Example the graphs suggest that the limaçon r c sin has an inner loop when c. Prove that this is true, and find the values of that correspond to the inner loop. (b) From Figure 9 it appears that the limaçon loses its dimple when c. Prove this. 56. Match the polar equations with the graphs labeled I VI. Give reasons for your choices. (Don t use a graphing device.) (a) r s, (b) r (c) r cos (d) r cos (e) r sin (f) r sin 6 5. I II III r _ 6, 9 8 Sketch the curve with the given polar equation. 9.. r r. r sin. r cos. r sin,. r cos 5. 6 r, 6. r ln, 7. r sin 8. r cos 5 9. r cos. r cos 6. r sin. r sin IV V VI
SECTIN. PLAR CRDINATES 69 57 6 Find the slope of the tangent line to the given polar curve at the point specified by the value of. 57. r sin, 58. r sin, 6. r cos, 6. r cos, 6 68 Find the points on the given curve where the tangent line is horizontal or vertical. 6. r cos 6 59. r, 6. r cos, 6. 67. r sin 68. r sin r sin 65. r cos 66. r e ; 8. A family of curves is given by the equations r c sin n, where c is a real number and n is a positive integer. How does the graph change as n increases? How does it change as c changes? Illustrate by graphing enough members of the family to support your conclusions. ; 8. A family of curves has polar equations r a cos a cos Investigate how the graph changes as the number a changes. In particular, you should identify the transitional values of a for which the basic shape of the curve changes. ; 8. The astronomer Giovanni Cassini (65 7) studied the family of curves with polar equations r c r cos c a 69. Show that the polar equation r a sin b cos, where ab, represents a circle, and find its center and radius. 7. Show that the curves r a sin and r a cos intersect at right angles. ; 7 76 Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. 7. r sin (nephroid of Freeth) 7. r s.8 sin (hippopede) 7. r e sin cos (butterfly curve) 7. r sin cos where a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and c. (Cassini thought that these curves might represent planetary orbits better than Kepler s ellipses.) Investigate the variety of shapes that these curves may have. In particular, how are a and c related to each other when the curve splits into two parts? 8. Let P be any point (except the origin) on the curve r f. If is the angle between the tangent line at P and the radial line P, show that tan r [Hint: bserve that drd in the figure.] 75. 76. r 5 sin6 r cos cos r=f( ) ÿ P ; 77. How are the graphs of r sin 6 and r sin related to the graph of r sin? In general, how is the graph of r f related to the graph of r f? ; 78. Use a graph to estimate the y-coordinate of the highest points on the curve r sin. Then use calculus to find the exact value. ; 79. (a) Investigate the family of curves defined by the polar equations r sin n, where n is a positive integer. How is the number of loops related to n? (b) What happens if the equation in part (a) is replaced by r sin n? 8. (a) Use Exercise 8 to show that the angle between the tangent line and the radial line is at every point on the curve r e. ; (b) Illustrate part (a) by graphing the curve and the tangent lines at the points where and. (c) Prove that any polar curve r f with the property that the angle between the radial line and the tangent line is a constant must be of the form r Ce k, where C and k are constants.
SECTIN. AREAS AND LENGTHS IN PLAR CRDINATES 65 FIGURE 8 r=+sin Formula 5 gives L y r y s sin d dr d d y We could evaluate this integral by multiplying and dividing the integrand by s sin, or we could use a computer algebra system. In any event, we find that the length of the cardioid is L 8. M s sin cos d. EXERCISES Find the area of the region that is bounded by the given curve and lies in the specified sector.. r,. r e,. r sin,. r ssin, 5 8 Find the area of the shaded region. 5. 6. 7. 9 Sketch the curve and find the area that it encloses.. r=œ r=+ sin r cos 8.. r=+cos r=sin 9. r cos. r cos r sin. r cos. r cos ; 5 6 Graph the curve and find the area that it encloses. 7 Find the area of the region enclosed by one loop of the curve. 7. r sin 8. r sin 5. r sin 6 6. r sin sin 9 9. r cos 5. r sin 6. r sin (inner loop). Find the area enclosed by the loop of the strophoid r cos sec. 8 Find the area of the region that lies inside the first curve and outside the second curve.. r cos, r. r sin, 5. r 8 cos, r 6. r sin, 7. r cos, r cos 8. r sin, 9 Find the area of the region that lies inside both curves. 9. r s cos,. r cos, r cos. r sin, r cos. r cos,. r sin,. r a sin, r b cos, a, b 5. Find the area inside the larger loop and outside the smaller loop of the limaçon r cos. 6. Find the area between a large loop and the enclosed small loop of the curve r cos. 7 Find all points of intersection of the given curves. 7. r sin, 8. r cos, r sin r sin r sin r cos r sin r sin r r sin 9. r sin, r. r cos, r sin. r sin, r sin. r sin, r cos
APPENDIX I ANSWERS T DD-NUMBERED EXERCISES A. 5. 7.5 y x, y x y (c) 8.5 x _, 7. (a) d sin r d cos 9. 7, 9 9 ),,. ab. e 5. r d x 7. s t dt.678 9. x. s s sin t cos t dt.67. s ln( s) s ln( s) 5. s e 8 ( 6. (a) (b) (c),,, 5 (, ),, _ (, s) _ 7. e e 8 9. 6.5 5. 55. (a) 5.5 5 6s, s t, (s, s) 5. (a) (i) (s, 7) (ii) (s, ) (b) (i), (ii), 5 7. 9. r= _, r= =_ = 6 r=. = 7 5 5 r= r= (b) 57. 9 x t e t se t t t t dt.5999 6 59. 5(7s 6) 6. 5a 6. 59. 65. 7. 5 (99s6 ) EXERCISES. N PAGE 67. (a) (b) 5,, _ _. s 5. Circle, center, radius 7. Circle, center (, ), radius 9. Horizontal line, unit above the x-axis. r sec. r cot csc 5. r c cos 7. (a) (b) x 9.. 6 = 5, =_ 6, 7,,, 5,,
A APPENDIX I ANSWERS T DD-NUMBERED EXERCISES. 5. 69. Center b, a, radius sa b 7. 7..6.5 _..8 _ 7. 9. = 6 5 = 8 75. 7 _.6 _.5 7 7.. = 5 6 = 6 5. 7. = = 77. By counterclockwise rotation through angle 6,, or about the origin 79. (a) A rose with n loops if n is odd and n loops if n is even (b) Number of loops is always n 8. For a, the curve is an oval, which develops a dimple as a l. When a, the curve splits into two parts, one of which has a loop. 7 (, ) (, ) EXERCISES. N PAGE 65 9. 5.. 5,. 8 s 5. 7. 9 9.. (, ) (6, ) 5. 55. (a) For c, the inner loop begins at and ends at ; for c, it begins at and ends at. sin c sin /c sin c sin c. 5. = 6 57. s 59. 6. 6. Horizontal at (s, ), (s, ) ; vertical at,,, 65. Horizontal at (, ),, [the pole], and (, 5) ; vertical at (, ), (, ), (, ) 67. Horizontal at,,, ; vertical at (, where sin ( s, ) s) ( s, ) s ( s s) 7. 9... 5. 5 s 7. 9... 5. 7. (, 6), (, 56), and the pole 9., where, 5,, 7 and, where,, 9, 8 8 9 7 s