## highschoolmom2010 2 years ago The Polish astronomer Nicolaus Copernicus devised a method for determining the sizes of the orbits of planets farther from the sun than Earth. His method involved noting the number of days between the times that a planet was in the positions labeled A and B in the diagram. Using this time and the number of days in each planet’s year, he calculated c and d.

1. highschoolmom2010

2. highschoolmom2010

3. highschoolmom2010

@ganeshie8

4. highschoolmom2010

@thomaster

5. ganeshie8

do u see a right triangle in that cosmic jungle ?

6. ganeshie8

|dw:1376506909669:dw|

7. highschoolmom2010

yes

8. ganeshie8

lets solve question A first :- say, x is the distance between Sun and Mars

9. ganeshie8

|dw:1376507051350:dw|

10. ganeshie8

|dw:1376507103333:dw|

11. ganeshie8

can u find that angle ? (look at the cosmic jungle pic again)

12. ganeshie8

that angle equals d-c = 103.8-55.2 = ?

13. ganeshie8

once u have that angle, u can find the x easily using an appropriate trig ratio

14. highschoolmom2010

48.68

15. ganeshie8

103.8 - 55.2 48.6

16. ganeshie8

its just 48.6

17. ganeshie8

how did u get 48.68 ?

18. highschoolmom2010

103.88accidentally pushed the |dw:1376507391898:dw|

19. ganeshie8

lol okie :)

20. highschoolmom2010

so now we know that <d =48.6 what do we do

21. ganeshie8

|dw:1376507462089:dw|

22. ganeshie8

u tell me, which ratio we can use here to find distance between Suna and Mars(x)

23. highschoolmom2010

|dw:1376507504635:dw|

24. ganeshie8

forget about million years. lets do the calculations in AUs which are simple

25. highschoolmom2010

we know hyp and adj SOH CAH TOA we use cos

26. highschoolmom2010

how do i do that

27. ganeshie8

yes cos ! and pls use AUs...

28. highschoolmom2010

how do i do that

29. ganeshie8

and the question is also asking u for AUs oly

30. ganeshie8

|dw:1376507639742:dw|

31. ganeshie8

hyp = x adj = 1 AU

32. highschoolmom2010

oh ok so i just put 1 AU not 93 million okie :D

33. ganeshie8

AU is just an unit, like meter or second. so dont bother about it much :)

34. ganeshie8

yes :)

35. ganeshie8

cos 48.6 = 1/x solve x

36. ganeshie8

do this, and do the Jupiter problem also same way. i need to go have dinner. brb.

37. highschoolmom2010

$\cos 48.6=\frac{ 1 }{ x }$ $x *\cos 48.6=\frac{ 1 }{ ?x }*x$ $x \cos 48.6=1$

38. highschoolmom2010

$\frac{x \cos 46.8 }{ \cos 46.8 }=\frac{ 1 }{ \cos 46.8 }$ $x=\frac{ 1 }{ \cos 46.8 }$

39. highschoolmom2010

do i do the inverse now

40. ganeshie8

just plugin that $$\large x = \frac{1}{\cos 48.6}$$

41. ganeshie8

you should get x = 1.5 AU approx.. try the Jupiter problem...

42. highschoolmom2010

|dw:1376587258601:dw|

43. highschoolmom2010

|dw:1376587399528:dw|

44. highschoolmom2010

$\cos 100.8=\frac{ 1 }{ x }$ $(x)* \cos 100.8=\frac{ 1 }{ x }*x$ $X \cos 100.8=1$ $\frac{ x \cos 100.8 }{ \cos 100.8 }=\frac{ 1 }{ \cos 100.8 }$

45. ganeshie8

angle is not 100.8

46. ganeshie8

|dw:1376587971593:dw|

47. ganeshie8

for jupitor c = 21.9, d = 100.8 angle = d-c = ?

48. highschoolmom2010

oh... 78.9

49. ganeshie8

yes do it again :)

50. highschoolmom2010

$\cos 78.9=\frac{ 1 }{ x }$ $(x)* \cos 78.9=\frac{ 1 }{ x }*(x)$ $x \cos 78.9=1$ $\frac{ x \cos 78.9 }{\cos 78.9 }=\frac{ 1 }{ \cos 78.9 }$ $x \approx 5.19$

51. highschoolmom2010

@ganeshie8

52. ganeshie8

perfect !

53. ganeshie8

now tell me this, did u get why we have been doing d-c for angle ?

54. highschoolmom2010

no not really

55. ganeshie8

look at the origianal cosmic pic, you wil understand

56. ganeshie8

observe that, $$d$$ is the angle moved by earth

57. ganeshie8

and $$c$$ is the anle movied by the other planet in same time

58. highschoolmom2010

i see that

59. ganeshie8

$$d-c$$ gives u the angle in that right triangle (visually u need to **see** it)

60. ganeshie8

|dw:1376589514631:dw|

61. ganeshie8

|dw:1376589569683:dw|

62. ganeshie8

that entire angle is $$d$$

63. highschoolmom2010

|dw:1376589610421:dw| right ?

64. ganeshie8

that part is $$c$$

65. ganeshie8

yes, right !

66. ganeshie8

So, whats the angle in that triangle ?

67. ganeshie8

|dw:1376589698138:dw|

68. highschoolmom2010

78.9

69. ganeshie8

uhh i mean in terms of $$d$$ and $$c$$

70. highschoolmom2010

im not sure what u mean

71. ganeshie8

|dw:1376589806634:dw|

72. highschoolmom2010

oh ok

73. ganeshie8

angle in the triangle equals $$d - c$$ you worked the problem correctly ! just wanted u to see why we did $$100.8-21.9$$

74. ganeshie8

hope it makes sense, the problem is easy actually - except for the fancy copernicus and planets words..

75. highschoolmom2010

yea the AU threw me off

76. ganeshie8

haha i thought so :)

77. highschoolmom2010

i think that was just uncalled for work

78. ganeshie8

hmm -_-

79. highschoolmom2010

ty @ganeshie8

80. ganeshie8

np :)