1. anonymous

2. jim_thompson5910

Start by drawing any generic triangle ABC |dw:1434070079074:dw|

3. jim_thompson5910

A = 43 degrees B = 62 degrees we can find C A+B+C = 180 43+62+C = 180 C = 75 |dw:1434070139361:dw|

4. jim_thompson5910

yes, BC = 22 and AB = unknown Let x = AB |dw:1434070198509:dw|

5. anonymous

yes I know that much

6. jim_thompson5910

now you use the law of sines sin(A)/a = sin(C)/c sin(43)/22 = sin(75)/x solve for x

7. anonymous

$\frac{ \sin43 }{ 22 } = \frac{ \sin75 }{ x }$

8. anonymous

I'm honestly not sure how to solve for x @jim_thompson5910

9. jim_thompson5910

using a calculator, you should find that sin(75) = 0.96592582628907 sin(43) = 0.6819983600625

10. jim_thompson5910

replace sin(43) and sin(75) with those approx values cross multiply and then solve for x

11. anonymous

ooh ok thank you

12. anonymous

this is what I got...

13. jim_thompson5910

you can also do it symbolically like this $\Large \frac{\sin(43)}{22}=\frac{\sin(75)}{x}$ $\Large \sin(43)*x=22*\sin(75)$ $\Large x=\frac{22*\sin(75)}{\sin(43)}$ $\Large x=???$ treating sin(43) and sin(75) as if they are variables

14. jim_thompson5910

desmos is in radian mode by default

15. jim_thompson5910

click the wrench icon and switch to degree mode

16. anonymous

Is there a place to put degrees in desmos?

17. jim_thompson5910

it's off to the right side

18. anonymous

oh thank you

19. anonymous

so the first one is D?

20. jim_thompson5910

yeah I get 31.1589725471071 which rounds to 31.2

21. anonymous

Thanks I think I can get the 2nd on my own

22. jim_thompson5910

np

23. anonymous

|dw:1434070810160:dw|

24. anonymous

|dw:1434070891025:dw|

25. anonymous

sorry I'm just using this to walk myself through it you can leave @jim_thompson5910

26. jim_thompson5910

alright, feel free to ask if you get stuck anywhere

27. anonymous

28. anonymous

I'm not sure what to use to find r... @jim_thompson5910

29. jim_thompson5910

you need qr to find r (with the law of sines) but before you can use the law of sines, you have to use the law of cosines to find qr

30. anonymous

ok...

31. jim_thompson5910

|dw:1434071160502:dw|

32. anonymous

$c^{2} = 7.5^{2} + 8.4^{2} - 2(7.5)(8.4)\cos(43)$

33. anonymous

like that?

34. jim_thompson5910

very good

35. jim_thompson5910

isolate c

36. anonymous

ok I'm working it out now...

37. anonymous

I got c = 7.54

38. jim_thompson5910

me too

39. anonymous

|dw:1434071506507:dw|

40. jim_thompson5910

now you can use the law of sines

41. anonymous

can you set it up for me like you did with the last one? I'm very bad at this...

42. jim_thompson5910

the law of sines works like this sin(A)/a = sin(B)/b = sin(C)/c |dw:1434071589360:dw|

43. jim_thompson5910

the letters pair up (upper and lower case) A pairs with a B with b C with c you'll have sin(A) pair with a, so that's how I got one fraction sin(A)/a same with sin(B)/b and sin(C)/c

44. jim_thompson5910

normally you don't know the full picture of the triangle so instead of saying sin(A)/a = sin(B)/b = sin(C)/c you would use sin(A)/a = sin(B)/b or sin(B)/b = sin(C)/c or sin(A)/a = sin(C)/c

45. anonymous

$\frac{ \sin43 }{ 7.5 } = \frac{ \sin7.5 }{ x }$

46. anonymous

so this works?

47. jim_thompson5910

I would use the full decimal expansion of qr instead of 7.5 or you can use a longer version, say qr = 7.5409365

48. jim_thompson5910

and the second fraction won't have sin(7.5)

49. jim_thompson5910

|dw:1434071876378:dw|

50. jim_thompson5910

|dw:1434071921347:dw|

51. anonymous

oh ok, so is this the right format (besides the larger decimal)?

52. jim_thompson5910

do you see how I have sin(r)/7.5 ?

53. anonymous

oooh ya I'll fix that

54. anonymous

$7.5= \frac{ 7.5*\sin x }{ \sin 43 }$

55. anonymous

I'm just screwing this up aren't I?

56. jim_thompson5910

you're doing great

57. jim_thompson5910

don't worry

58. anonymous

ok but how do I isolate x?

59. jim_thompson5910

First cross multiply. Then isolate sin(x). Finally use arcsine to fully isolate x $\Large \frac{\sin(43)}{7.5409365}=\frac{\sin(x)}{7.5}$ $\Large 7.5*\sin(43)=7.5409365*\sin(x)$ $\Large 7.5409365*\sin(x) = 7.5*\sin(43)$ $\Large \sin(x)=\frac{7.5*\sin(43)}{7.5409365}$ $\Large x=???$

60. anonymous

OOOH thank you so much!

61. jim_thompson5910

what result do you get

62. anonymous

um... I got lost after this far

63. anonymous

I'm not sure what to do with the sin x to find x

64. jim_thompson5910

the top 7.5 should use more decimal digits (say 4 or 5)

65. jim_thompson5910

oops I meant the bottom 7.5

66. anonymous

didn't change much

67. jim_thompson5910

oh wait, I just realized that qr isn't 7.5 you were in radian mode on accident (I was too)

68. jim_thompson5910

no wonder I'm not getting the answer choices

69. anonymous

woooow XD ok then what is it??

70. jim_thompson5910

this is the length of qr

71. jim_thompson5910

QR = 5.887226 approximately |dw:1434072742525:dw|

72. anonymous

here's what I got

73. jim_thompson5910

$\Large \frac{\sin(43)}{5.887226}=\frac{\sin(x)}{7.5}$ $\Large 7.5*\sin(43)=5.887226*\sin(x)$ $\Large 5.887226*\sin(x) = 7.5*\sin(43)$ $\Large \sin(x)=\frac{7.5*\sin(43)}{5.887226}$ $\Large x=???$

74. anonymous

oh oops I changed the other 7.5 XD

75. anonymous

there we go!

76. jim_thompson5910

now evaluate the arcsine of that number

77. jim_thompson5910

78. anonymous

Ta Da!

79. jim_thompson5910

you nailed it

80. anonymous

so its D?

81. jim_thompson5910

both seem to be D, yes I guess the makers weren't too creative

82. anonymous

haha well thank you SO much! *hugs*

83. jim_thompson5910

you're welcome