## SugarRainbow 3 years ago find the value of the expression if sin beta= 0.45, find cos(pi/2-beta)

1. SugarRainbow

do i just do pi/2-0.45?

2. lgbasallote

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

3. lgbasallote

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

4. SugarRainbow

no idk what you just did

5. lgbasallote

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

6. lgbasallote

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

7. SugarRainbow

yeah but whats beta?

8. Algebraic!

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9. lgbasallote

ahh we'll get there

10. Algebraic!

11. lgbasallote

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

12. Algebraic!

13. lgbasallote

i was talking to sugarrainbow

14. Skaematik

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

15. SugarRainbow

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. lgbasallote

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

17. lgbasallote

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

18. SugarRainbow

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

19. SugarRainbow

and how come we add cos+sin?

20. Skaematik

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. tanjung

cos(pi/2-beta) = sin beta

22. SugarRainbow

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. SugarRainbow

when i did pi/2 i got 0.5

24. tanjung

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

25. SugarRainbow

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

26. tanjung

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

27. Algebraic!

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28. Algebraic!

cotangent of beta = 1/-5.32

29. SugarRainbow

how did you get -5.32

30. Algebraic!

see beta?

31. Algebraic!

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

32. Algebraic!

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

33. SugarRainbow

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

34. Algebraic!

what problem are you working on?

35. SugarRainbow

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

36. SugarRainbow

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

37. Algebraic!

sure, look carefully at the sketch plz

38. Algebraic!

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39. Algebraic!

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

40. Algebraic!

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41. Algebraic!

opposite over hypotenuse is sine

42. SugarRainbow

okay that makes sense

43. Algebraic!

cool:)

44. SugarRainbow

but now how do i get cos?

45. Algebraic!

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

46. Algebraic!

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

47. Algebraic!

so that angle must be pi/2 - Beta

48. SugarRainbow

does this have anything to do with the unit circle

49. Algebraic!

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

50. Algebraic!

yeah, it all is the same thing..

51. Algebraic!

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

52. Algebraic!

|dw:1347257807464:dw|

53. Algebraic!

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

54. Algebraic!

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

55. Algebraic!

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. Algebraic!

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. Algebraic!

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58. Algebraic!

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59. Algebraic!

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60. SugarRainbow

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

61. SugarRainbow

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

62. Algebraic!

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. Algebraic!

pi/2 != 1

64. Algebraic!

pi/2 = pi/2

65. SugarRainbow

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. Algebraic!

ah the hypotenuse =1 you mean?

67. SugarRainbow

yeah

68. Algebraic!

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

69. Algebraic!

70. Algebraic!

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. SugarRainbow

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

72. Algebraic!

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. SugarRainbow

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. Algebraic!

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. SugarRainbow

but what's pi/2?

76. Algebraic!

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

77. Algebraic!

pi/2 is 90 deg.

78. Algebraic!

1/4 the way around a circle.

79. SugarRainbow

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. Algebraic!

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

81. Algebraic!

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82. Algebraic!

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83. SugarRainbow

okay

84. Algebraic!