amorfide
  • amorfide
I need someone to explain the CAST diagram to me please? I need to know how to find all possible angles between -360 and 360 for sin cosin and tan
Mathematics
schrodinger
  • schrodinger
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asnaseer
  • asnaseer
It is explained quite well in this video: http://www.youtube.com/watch?v=FpjWXKbCC3w
amorfide
  • amorfide
@asnaseer thank you but what i need is lets say we have sinx=0.5 x=sin^-1(0.5) i need to know how to get all possible angles
asnaseer
  • asnaseer
ok, what you need to do in this specific case is note that the primary solution to:\[\sin(x)=0.5\]is:\[x=30^0=\frac{\pi}{6}\]Now, using CAST we also know that:\[\sin(x)=\sin(\pi-x)\]therefore:\[x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}\]Next note that the sine curve repeats itself every \(2\pi\), so the general solution would be:\[x=\frac{\pi}{6}+2n\pi\text{ for }n=-\infty,...,-1,0,1,...,\infty\]and:\[x=\frac{5\pi}{6}+2n\pi\text{ for }n=-\infty,...,-1,0,1,...,\infty\]

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asnaseer
  • asnaseer
I hope that makes sense?
amorfide
  • amorfide
it made sense until you brought n into this... what is n exactly?
asnaseer
  • asnaseer
n is just an integer that can take on any value from \(-\infty\) to \(+\infty\). e.g. if n=0, we get:\[x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}\]All n does is add (or subtract) \(2\pi\) to this answer depending on whether n is positive or negative.
asnaseer
  • asnaseer
that should say "add (or subtract) multiples of \(2\pi\)..."
amorfide
  • amorfide
but i have never seen a question give me a value of n? so how would i find values for cosin? cos(x)=0.5 x=cos^-1(0.5)
asnaseer
  • asnaseer
If you haven't been introduced to this general form of the answer, then ignore it for now. So the solutions to your first question would just be:\[x=\frac{\pi}{6}\text{ or }x=\frac{5\pi}{6}\] Now for the second question, first tel me what you think the primary solution is to:\[\cos(x)=0.5\]
asnaseer
  • asnaseer
Hang on - you are supposed to find solutions between -360 and +360, correct?
amorfide
  • amorfide
yes -360 to 360 sorry helping others too
amorfide
  • amorfide
i know how to convert to radians
asnaseer
  • asnaseer
So first draw these two solutions we found as follows: |dw:1349974770993:dw|
asnaseer
  • asnaseer
So then the negative angles that also satisfy the first equation will be: |dw:1349974926971:dw|
asnaseer
  • asnaseer
Does that make sense?
amorfide
  • amorfide
the -210 makes sense, the -330 does not
asnaseer
  • asnaseer
-360 + 150 = -210 -360 + 30 = -330
asnaseer
  • asnaseer
|dw:1349975262412:dw|
asnaseer
  • asnaseer
|dw:1349975300819:dw|
asnaseer
  • asnaseer
Does that help?
amorfide
  • amorfide
okay so sin is fine cosine?
asnaseer
  • asnaseer
Exactly the same principles
asnaseer
  • asnaseer
|dw:1349975392520:dw|
amorfide
  • amorfide
thank you! you are like my hero i can do most of the complicated things but this was always just too hard :P i understand it better now, thank you!
asnaseer
  • asnaseer
you are more than welcome my friend! I'm glad I was able to explain it to you. :)

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