## anonymous 4 years ago find the value of the expression if sin beta= 0.45, find cos(pi/2-beta)

1. anonymous

do i just do pi/2-0.45?

2. anonymous

sin beta is 0.45...but beta is not 0.45 does that make sense?

3. anonymous

$\Large \cos (\frac \pi 2 - \beta) \implies \cos \frac \pi 2 \cos \beta + \sin \frac \pi 2 \sin \beta$ are you familiar with this?

4. anonymous

no idk what you just did

5. anonymous

it's a trig formula $\LARGE \cos (A - B ) \implies \cos A \cos B + \sin A \sin B$

6. anonymous

in this case, our A is pi/2 and B is beta make sense?

7. anonymous

yeah but whats beta?

8. anonymous

|dw:1347254638009:dw|

9. anonymous

ahh we'll get there

10. anonymous

11. anonymous

so you get how $\cos(\frac \pi 2 - \beta) = \cos \frac \pi 2 \cos \beta + \sin \frac \pi 2 \sin \beta$

12. anonymous

answer is clearly .45

13. anonymous

i was talking to sugarrainbow

14. anonymous

Is it just not cos(pi/2 - sin^-1(0.45))?

15. anonymous

umm kay i don't get algebraic's picture and i not sure how igbasallote's formula works if we don't know what beta is

16. anonymous

it works because $\cos \frac \pi 2 = 0$ $\sin \frac \pi 2 = 1$

17. anonymous

so if you substitute you get $\implies (0) \cos \beta + (1) \sin \beta$

18. anonymous

what about what Skaematik did? can't i just do that?

19. anonymous

and how come we add cos+sin?

20. anonymous

This is a formula that you will need to memorise. Its something that you can use to manipulate trignometric equations to find an answer cos(A−B)⟹cosAcosB+sinAsinB Therefore if you use it in this case, like lgbasallote said cos(π2−β)=cosπ/2cosβ+sinπ/2sinβ Plug these values into your calculator cosπ/2 sinπ/2 And you will find they equal to 0 and 1 Therefore cos(π2−β)=0+1*sinβ Sinβ =0.45 Therefore the answer is 0.45

21. anonymous

cos(pi/2-beta) = sin beta

22. anonymous

okay that makes more sense but what would the formula be if i had tangent(pi/2-beta)=-5.32 and i have to find cot?

23. anonymous

when i did pi/2 i got 0.5

24. anonymous

tangent (pi/2-beta) still in the first kuadrant, so impossible has a negative value and tangent (pi/2-beta)=cotangent beta

25. anonymous

okay so is that what cosine is or how do we get that

26. anonymous

maybe ur problem is if given tangent(pi/2-beta)=5.32, find cot beta? cot beta = tan(pi/2-beta) = 5.32

27. anonymous

|dw:1347256883382:dw|

28. anonymous

cotangent of beta = 1/-5.32

29. anonymous

how did you get -5.32

30. anonymous

see beta?

31. anonymous

what's the tangent? opp = -5.32 adjacent =1

32. anonymous

tan(beta) = -5.32 and cot(beta) = 1/-5.32

33. anonymous

but how did you get that if we have sin and cos?

34. anonymous

what problem are you working on?

35. anonymous

oh i'm sorry i was still looking at the first one

36. anonymous

can we just go back to the first one cuz i still don't understand it

37. anonymous

sure, look carefully at the sketch plz

38. anonymous

|dw:1347257483078:dw|

39. anonymous

do you agree? sin(beta) = .45/1

40. anonymous

|dw:1347257532660:dw|

41. anonymous

opposite over hypotenuse is sine

42. anonymous

okay that makes sense

43. anonymous

cool:)

44. anonymous

but now how do i get cos?

45. anonymous

now: |dw:1347257619039:dw| what's that new angle?

46. anonymous

the whole angle from the x axis to the y axis is 90 or pi/2 ....

47. anonymous

so that angle must be pi/2 - Beta

48. anonymous

does this have anything to do with the unit circle

49. anonymous

now look at the cosine of that angle: |dw:1347257708887:dw|

50. anonymous

yeah, it all is the same thing..

51. anonymous

see that triangle. one side is .45 the hypotenuse is 1 the other side, no one cares what it is!

52. anonymous

|dw:1347257807464:dw|

53. anonymous

the cosine of that new angle is clearly .45 /1 .45 is adjacent, 1 is the hypotenuse.

54. anonymous

so cos(pi/2 -Beta) is .45

55. anonymous

it's what @tanjung said : cos(pi/2-beta) = sin beta but I showed it graphical so you could believe! you can't really argue with that picture.

56. anonymous

basically if two angles are complimentary, the cosine of one angle is the sin of the other.... ie the side that is opposite on one triangle *must be* the side that is adjacent on the other triangle...

57. anonymous

|dw:1347258175266:dw|

58. anonymous

|dw:1347258227158:dw|

59. anonymous

|dw:1347258316959:dw|

60. anonymous

umm okay is it also because its sin and cos that its the same?

61. anonymous

real quick, since pi/2=1 then would i just divide .45(sine beta) by 1 to get cos?

62. anonymous

yeah, I guess I get what you're saying... it comes from the definition of sin as opp/hyp and cosine as adj/hyp when you draw those sides for two angles which are complimentary they make a rectangle

63. anonymous

pi/2 != 1

64. anonymous

pi/2 = pi/2

65. anonymous

umm okay so thinking of it in terms of a rectangle that means cos is the same as sin because the two sides are equal?|dw:1347258695721:dw|

66. anonymous

ah the hypotenuse =1 you mean?

67. anonymous

yeah

68. anonymous

cosine isn't the same as sine.... that's not what I said, but yeah you're starting to get it, I think...

69. anonymous

your pic is accurate

70. anonymous

you see that, in your pic: |dw:1347258903616:dw| those two angles aren't the same? they do add up to give pi/2 (90 degrees) though so the cosine of one angle = the sine of the other:)

71. anonymous

oh so its only the same cuz it the angle adds up to 90?

72. anonymous

no, it works elsewhere on the circle. but just try to grasp this 1st quadrant sketch for now... then you can move on to more advanced stuff :)

73. anonymous

umm okay i do get how you got hyp=1 and opp=.45 but what bout the angles how did you get that they'd = 90?

74. anonymous

well remember, because if Beta is one of the angles and the other is pi/2 - Beta then Beta + (pi/2 - Beta) = pi/2

75. anonymous

but what's pi/2?

76. anonymous

so we're automatically talking about two angles that add up to pi/2.

77. anonymous

pi/2 is 90 deg.

78. anonymous

1/4 the way around a circle.

79. anonymous

oh i just put it in my calc and i see how you got 90 but how does that necessarily help us with this because a bunch of different combinations of angles can add up to 90

80. anonymous

that's good actually. that means it works no matter what Beta is. It's general. For any Beta.

81. anonymous

|dw:1347259786709:dw|

82. anonymous

|dw:1347259882928:dw|

83. anonymous

okay

84. anonymous

still talking about first quadrant stuff at the moment..

85. anonymous

anyway hope that cleared it up. think it over a bit, you'll see why it has to be true.