第10章三角法のさらなる応用Further Applications of Trigonometry

Summary of Curves曲線のまとめ

We have explored a number of seemingly complex polar curves in this section. Figure 20 and Figure 21 summarize the graphs and equations for each of these curves.

この節では、一見込み入った極の曲線をいくつも調べた。図20と図21に、この曲線それぞれのグラフと式をまとめる。

Four graphs side by side - a summary. (A) is a circle: r=asin(theta) or r=acos(theta). (B) is a cardioid: r= a + or - bcos(theta), or r = a + or - b sin(theta). a>0, b>0, a/b=1. (C) is one-loop limaçons. r= a + or - bcos(theta), or r= a + or - bsin(theta). a>0, b>0, 1<a/b<2. (D) is inner-loop limaçons. R = a + or - bcos(theta), or r = a + or - bsin(theta). A>0, b>0, a<b.

Four graphs side by side - a summary. (A) is lemniscates. R^2 = a^2cos(2theta), or r^2=a^2sin(2theta). a is not equal to 0. (B) is a rsose curve (n even). R = acos(ntheta), or r=asin(ntheta). N is even, and there are 2n petals. (C) is a rose curve (n odd). R = acos(ntheta), or r=asin(theta). N is odd, and there are n petals. (D) is an Archimedes's spiral. R=theta, and theta >=0.

Media 教材

Access these online resources for additional instruction and practice with graphs of polar coordinates.

極座標のグラフについての説明と練習をさらに求めるなら、次のオンライン教材にあたるとよい。

• Graphing Polar Equations Part 1• Graphing Polar Equations Part 2• Animation: The Graphs of Polar Equations• Graphing Polar Equations on the TI-84

・極方程式のグラフを描く第1部・極方程式のグラフを描く第2部・動画: 極方程式のグラフ・TI-84 で極方程式のグラフを描く

Verbal言葉で答える問題

1. Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

1. 極のグラフの三つの種類の対称性を述べ、デカルト平面の対称性と比べよ。

2. Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x-axis, y-axis, and origin?

2. 極のグラフの三つの種類の対称性のうち、x軸・y軸・原点に関する対称性に当たるのはどれか。

3. What are the steps to follow when graphing polar equations?

3. 極方程式のグラフを描くとき、どんな手順をたどるか。

4. Describe the shapes of the graphs of cardioids, limaçons, and lemniscates.

4. カージオイド・リマソン・レムニスケートのグラフの形を述べよ。

5. What part of the equation determines the shape of the graph of a polar equation?

5. 極方程式のグラフの形は、式のどの部分で決まるか。

Graphicalグラフの問題

For the following exercises, test the equation for symmetry.

次の各問について、式の対称性を調べよ。

6. $r = 5 \cos 3 θ$

7. $r = 3 - 3 \cos θ$

8. $r = 3 + 2 \sin θ$

9. $r = 3 \sin 2 θ$

10. $r = 4$

11. $r = 2 θ$

12. $r = 4 \cos \frac{θ}{2}$

13. $r = \frac{2}{θ}$

14. $r = 3 \sqrt{1 - \cos^{2} θ}$

15. $r = \sqrt{5 \sin 2 θ}$

For the following exercises, graph the polar equation. Identify the name of the shape.

次の各問について、極方程式のグラフを描け。その形の名前を答えよ。

16. $r = 3 \cos θ$

17. $r = 4 \sin θ$

18. $r = 2 + 2 \cos θ$

19. $r = 2 - 2 \cos θ$

20. $r = 5 - 5 \sin θ$

21. $r = 3 + 3 \sin θ$

22. $r = 3 + 2 \sin θ$

23. $r = 7 + 4 \sin θ$

24. $r = 4 + 3 \cos θ$

25. $r = 5 + 4 \cos θ$

26. $r = 10 + 9 \cos θ$

27. $r = 1 + 3 \sin θ$

28. $r = 2 + 5 \sin θ$

29. $r = 5 + 7 \sin θ$

30. $r = 2 + 4 \cos θ$

31. $r = 5 + 6 \cos θ$

32. $r^{2} = 36 \cos (2 θ)$

33. $r^{2} = 10 \cos (2 θ)$

34. $r^{2} = 4 \sin (2 θ)$

35. $r^{2} = 10 \sin (2 θ)$

36. $r = 3 sin (2 θ)$

37. $r = 3 cos (2 θ)$

38. $r = 5 sin (3 θ)$

39. $r = 4 sin (4 θ)$

40. $r = 4 sin (5 θ)$

41. $r = - θ$

42. $r = 2 θ$

43. $r = - 3 θ$

Technology機器を使う問題

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

次の各問について、グラフ電卓を使って極方程式のグラフの概形を描け。

44. $r = \frac{1}{θ}$

45. $r = \frac{1}{\sqrt{θ}}$

46. $r = 2 \sin θ \tan θ,$ a cissoid

46. $r = 2 \sin θ \tan θ$ シソイド

47. $r = 2 \sqrt{1 - \sin^{2} θ}$ , a hippopede

47. $r = 2 \sqrt{1 - \sin^{2} θ}$ 、ヒッポペデ

48. $r = 5 + \cos (4 θ)$

49. $r = 2 - \sin (2 θ)$

50. $r = θ^{2}$

51. $r = θ + 1$

52. $r = θ \sin θ$

53. $r = θ \cos θ$

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $[0, 4 π]$ and then explain the differences shown in the graphs.

次の各問について、グラフ作成ツールを使って、定義域 $[0, 4 π]$ で各組の極方程式のグラフを描き、グラフに見える違いを説明せよ。

54. $r = θ, r = - θ$

55. $r = θ, r = θ + \sin θ$

56. $r = \sin θ + θ, r = \sin θ - θ$

57. $r = 2 \sin (\frac{θ}{2}), r = θ \sin (\frac{θ}{2})$

58. $r = \sin (\cos (3 θ)) r = \sin (3 θ)$

59. On a graphing utility, graph $r = \sin (\frac{16}{5} θ)$ on $[0$ , $4 π]$ , $[0$ , $8 π]$ , $[0$ , $12 π]$ , and $[0$ , $16 π] .$ Describe the effect of increasing the width of the domain.

59. グラフ作成ツールで、 $[0$ 、 $4 π]$ 、 $[0$ 、 $8 π]$ 、 $[0$ 、 $12 π]$ 、 $[0$ 、 $16 π]$ で $r = \sin (\frac{16}{5} θ)$ のグラフを描け。定義域の幅を広げる効果を述べよ。

60. On a graphing utility, graph and sketch $r = \sin θ + {(\sin (\frac{5}{2} θ))}^{3}$ on $[0, 4 π] .$

60. グラフ作成ツールで、 $[0, 4 π]$ の $r = \sin θ + {(\sin (\frac{5}{2} θ))}^{3}$ のグラフを描き、概形を描け。

61. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

61. グラフ作成ツールで、各極方程式のグラフを描け。グラフに見える似ている点と違う点を説明せよ。

\begin{array}{l} r_{1} = 3 \sin (3 θ) \\ r_{2} = 2 \sin (3 θ) \\ r_{3} = \sin (3 θ) \end{array}

62. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

62. グラフ作成ツールで、各極方程式のグラフを描け。グラフに見える似ている点と違う点を説明せよ。

\begin{array}{l} r_{1} = 3 + 3 \cos θ \\ r_{2} = 2 + 2 \cos θ \\ r_{3} = 1 + \cos θ \end{array}

63. On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

63. グラフ作成ツールで、各極方程式のグラフを描け。グラフに見える似ている点と違う点を説明せよ。

\begin{array}{l} r_{1} = 3 θ \\ r_{2} = 2 θ \\ r_{3} = θ \end{array}

Extensions発展問題

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

次の各問について、各極方程式を同じ極の座標軸に描き、交点を求めよ。

64. $r_{1} = 3 + 2 \sin θ, r_{2} = 2$

65. $r_{1} = 6 - 4 \cos θ, r_{2} = 4$

66. $r_{1} = 1 + \sin θ, r_{2} = 3 \sin θ$

67. $r_{1} = 1 + \cos θ, r_{2} = 3 \cos θ$

68. $r_{1} = \cos (2 θ), r_{2} = \sin (2 θ)$

69. $r_{1} = \sin^{2} (2 θ)$ , $r_{2} = 1 - \cos (4 θ)$

70. $r_{1} = \sqrt{3}, r_{2} = 2 \sin (θ)$

71. $r_{1}^{2} = \sin θ, r_{2}^{2} = \cos θ$

72. $r_{1} = 1 + \cos θ$ , $r_{2} = 1 - \sin θ$