anonymous
  • anonymous
find the value of the expression if sin beta= 0.45, find cos(pi/2-beta)
Mathematics
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anonymous
  • anonymous
find the value of the expression if sin beta= 0.45, find cos(pi/2-beta)
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
do i just do pi/2-0.45?
lgbasallote
  • lgbasallote
sin beta is 0.45...but beta is not 0.45 does that make sense?
lgbasallote
  • 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?

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anonymous
  • anonymous
no idk what you just did
lgbasallote
  • lgbasallote
it's a trig formula \[\LARGE \cos (A - B ) \implies \cos A \cos B + \sin A \sin B\]
lgbasallote
  • lgbasallote
in this case, our A is pi/2 and B is beta make sense?
anonymous
  • anonymous
yeah but whats beta?
anonymous
  • anonymous
|dw:1347254638009:dw|
lgbasallote
  • lgbasallote
ahh we'll get there
anonymous
  • anonymous
already did.
lgbasallote
  • lgbasallote
so you get how \[\cos(\frac \pi 2 - \beta) = \cos \frac \pi 2 \cos \beta + \sin \frac \pi 2 \sin \beta\]
anonymous
  • anonymous
answer is clearly .45
lgbasallote
  • lgbasallote
i was talking to sugarrainbow
anonymous
  • anonymous
Is it just not cos(pi/2 - sin^-1(0.45))?
anonymous
  • 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
lgbasallote
  • lgbasallote
it works because \[\cos \frac \pi 2 = 0\] \[\sin \frac \pi 2 = 1\]
lgbasallote
  • lgbasallote
so if you substitute you get \[\implies (0) \cos \beta + (1) \sin \beta\]
anonymous
  • anonymous
what about what Skaematik did? can't i just do that?
anonymous
  • anonymous
and how come we add cos+sin?
anonymous
  • 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
anonymous
  • anonymous
cos(pi/2-beta) = sin beta
anonymous
  • 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?
anonymous
  • anonymous
when i did pi/2 i got 0.5
anonymous
  • anonymous
tangent (pi/2-beta) still in the first kuadrant, so impossible has a negative value and tangent (pi/2-beta)=cotangent beta
anonymous
  • anonymous
okay so is that what cosine is or how do we get that
anonymous
  • anonymous
maybe ur problem is if given tangent(pi/2-beta)=5.32, find cot beta? cot beta = tan(pi/2-beta) = 5.32
anonymous
  • anonymous
|dw:1347256883382:dw|
anonymous
  • anonymous
cotangent of beta = 1/-5.32
anonymous
  • anonymous
how did you get -5.32
anonymous
  • anonymous
see beta?
anonymous
  • anonymous
what's the tangent? opp = -5.32 adjacent =1
anonymous
  • anonymous
tan(beta) = -5.32 and cot(beta) = 1/-5.32
anonymous
  • anonymous
but how did you get that if we have sin and cos?
anonymous
  • anonymous
what problem are you working on?
anonymous
  • anonymous
oh i'm sorry i was still looking at the first one
anonymous
  • anonymous
can we just go back to the first one cuz i still don't understand it
anonymous
  • anonymous
sure, look carefully at the sketch plz
anonymous
  • anonymous
|dw:1347257483078:dw|
anonymous
  • anonymous
do you agree? sin(beta) = .45/1
anonymous
  • anonymous
|dw:1347257532660:dw|
anonymous
  • anonymous
opposite over hypotenuse is sine
anonymous
  • anonymous
okay that makes sense
anonymous
  • anonymous
cool:)
anonymous
  • anonymous
but now how do i get cos?
anonymous
  • anonymous
now: |dw:1347257619039:dw| what's that new angle?
anonymous
  • anonymous
the whole angle from the x axis to the y axis is 90 or pi/2 ....
anonymous
  • anonymous
so that angle must be pi/2 - Beta
anonymous
  • anonymous
does this have anything to do with the unit circle
anonymous
  • anonymous
now look at the cosine of that angle: |dw:1347257708887:dw|
anonymous
  • anonymous
yeah, it all is the same thing..
anonymous
  • anonymous
see that triangle. one side is .45 the hypotenuse is 1 the other side, no one cares what it is!
anonymous
  • anonymous
|dw:1347257807464:dw|
anonymous
  • anonymous
the cosine of that new angle is clearly .45 /1 .45 is adjacent, 1 is the hypotenuse.
anonymous
  • anonymous
so cos(pi/2 -Beta) is .45
anonymous
  • 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.
anonymous
  • 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...
anonymous
  • anonymous
|dw:1347258175266:dw|
anonymous
  • anonymous
|dw:1347258227158:dw|
anonymous
  • anonymous
|dw:1347258316959:dw|
anonymous
  • anonymous
umm okay is it also because its sin and cos that its the same?
anonymous
  • anonymous
real quick, since pi/2=1 then would i just divide .45(sine beta) by 1 to get cos?
anonymous
  • 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
anonymous
  • anonymous
pi/2 != 1
anonymous
  • anonymous
pi/2 = pi/2
anonymous
  • 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|
anonymous
  • anonymous
ah the hypotenuse =1 you mean?
anonymous
  • anonymous
yeah
anonymous
  • anonymous
cosine isn't the same as sine.... that's not what I said, but yeah you're starting to get it, I think...
anonymous
  • anonymous
your pic is accurate
anonymous
  • 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:)
anonymous
  • anonymous
oh so its only the same cuz it the angle adds up to 90?
anonymous
  • 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 :)
anonymous
  • 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?
anonymous
  • 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
anonymous
  • anonymous
but what's pi/2?
anonymous
  • anonymous
so we're automatically talking about two angles that add up to pi/2.
anonymous
  • anonymous
pi/2 is 90 deg.
anonymous
  • anonymous
1/4 the way around a circle.
anonymous
  • 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
anonymous
  • anonymous
that's good actually. that means it works no matter what Beta is. It's general. For any Beta.
anonymous
  • anonymous
|dw:1347259786709:dw|
anonymous
  • anonymous
|dw:1347259882928:dw|
anonymous
  • anonymous
okay
anonymous
  • anonymous
still talking about first quadrant stuff at the moment..
anonymous
  • anonymous
anyway hope that cleared it up. think it over a bit, you'll see why it has to be true.

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