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