Proving cos(cos-1(x)) Lies in [-1,1]

[SteliosVas]

Homework Statement

Show cos(cos^-1(x) lies in range [-1,1]

Homework Equations

I know the derivative of cos(x) is arcos(x).

The Attempt at a Solution

D/dx(cos(x)) = arcos(x)

However i am stuck how to prove cos(arcos(x)) has a domain of [-1,1]..

 

[mamort]

I think it is taken in context of a trig function in which cosine is described by a circle with a radius equal to one. Then since the cos and it's inverse cancel each other out leave the answer is as follows

cos(cos-1(x)) = x

This would simply mean that the line would run along the x axis, in terms of trig terminology this would be [-1,1].

 

[rtsswmdktbmhw]

I know the derivative of cos(x) is arcos(x).

You might want to check again.

 

[mamort]

Sorry I didn't read the rest, arcos(x) is the inverse of cos(x) not the derivative. Derivative is as follows

d/dx cos(x) = -sin(x)
d/dx sin(x) = cos(x)

This is because sin(x) and cos(x) are essentially the same function except the phase between the two is pi/2 (90 degrees or 1/4 revolution of a full circle), hence one can be derived by the other.

 

[SteamKing]

Homework Equations

I know the derivative of cos(x) is arcos(x).

You should forget this bit of mathematical misinformation immediately.

The derivative of cos(x) = -sin(x)

Here is a list of other derivatives:

http://en.wikipedia.org/wiki/Derivative

You should check what you 'know' about derivatives of other functions against this list.

The 'arccos (x)' is the inverse of cosine (x), not the derivative, which has a specific meaning in mathematics.

arccos(cosine(x)) = x

 

[RUber]

For what values of x is $\cos^{-1} x$ defined ( domain ).
X can only be in that interval, other than that--
$f(f^{-1}(x))=x$ all the time by definition.

 

[RUber]

Hint:arccos (x)=y means cos(y) = x.

 

[vela]

The 'arccos (x)' is the inverse of cosine (x), not the derivative, which has a specific meaning in mathematics.

arccos(cosine(x)) = x

This should really be the other way around: cos(arccos(x)) = x. It's possible for arccos(cos(x)) to differ from x, when x isn't in the range of the arccosine function.

 

[Ray Vickson]

Homework Statement

Show cos(cos^-1(x) lies in range [-1,1]

Homework Equations

I know the derivative of cos(x) is arcos(x).

The Attempt at a Solution

D/dx(cos(x)) = arcos(x)

However i am stuck how to prove cos(arcos(x)) has a domain of [-1,1]..

Have you looked at a graph of $y = \cos(\theta)$ over a broad range of $\theta$? I suggest you do that first, before trying to use fancy identities or derivatives and the like. Then, just remember what $\arccos(x)$ stands for.

 

[GFauxPas]

In the problem you're analyzing the image of cos(arccos(x)).
But then you say you're trying to figure out the domain of cos(arccos(x)). Which is it?

 

[RUber]

Before you can know the range (image) of a function, you must know which inputs would be valid.
For example, you could look at cos(arccos(x)) and say, hey, that's a function and its inverse, so the output is just x.
Then, you would say, hey, that doesn't make sense, since I know cosine has a range between -1 and 1...
So, you need to know the possible input values for x (domain of arccos(x)) in order to properly resolve this apparent contradiction.

 

[Mark44]

Let's hold off on any more replies until the OP comes back.