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En
 one year ago
please help!
prove that!
arccos (x)= πarccos x
En
 one year ago
please help! prove that! arccos (x)= πarccos x

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beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1Remember the definition of arccos x. It's the _angle_ whose cosine is x. So this identity says that two angles, one of whose cosine is x, and another whose cosine is (x) add up to pi (that is 180 degrees).

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1A figure of the unit circle might illustrate it better: dw:1441508969839:dw

En
 one year ago
Best ResponseYou've already chosen the best response.0@beginnersmind after that ? how am i going to start to prove it?

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1I would probably write it up in words. Maybe there's a way to do it by manipulating the identity, but I don't see it.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1Ummmm this is what I would do :) \[\large\rm \arccos (x)=\color{orangered}{π\arccos x}\]Let's call this right side something... like... y.\[\large\rm y=\pi\arccos x\]Let's subtract pi from each side, then multiply by 1, giving us,\[\large\rm \piy=\arccos x\]Take the inverse,\[\large\rm \cos(\piy)=x\]So if you subtract an angle from pi within cosine, that's the same as flipping it over the xaxis. Therefore:\[\large\rm \cos(\piy)=\cos(y)=x\]Inverse again,\[\large\rm y=\arccos(x)\]Relate this back to the original equation. Since the right side is arccos(x) and the left side is arccos(x), proofed! :O Maybe not the most straight forward approach :\ but whatev

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1From this step:\[\large\rm \cos(\piy)=x\]If you would like a better justification, you can apply your cosine angle difference identity:\[\large\rm \cos(\alpha\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta\]\[\large\rm \cos(\piy)=\cos \pi \cos y+\sin \pi \sin y\]Which will simplify down to cos y

beginnersmind
 one year ago
Best ResponseYou've already chosen the best response.1I think you should be able to use cos(x) = cos(pix). It's more basic than the subtraction formula.

zepdrix
 one year ago
Best ResponseYou've already chosen the best response.1ya i was hoping so :) wasn't sure how basic it was though hehe

En
 one year ago
Best ResponseYou've already chosen the best response.0thanks a lot! I'm gonna go and try to understand it :)))

En
 one year ago
Best ResponseYou've already chosen the best response.0@zepdrix and @beginnersmind i understood it now :)
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