1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of Arccos

  1. Oct 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Show cos(cos-1(x) lies in range [-1,1]
    2. Relevant equations

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

    3. 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]..
  2. jcsd
  3. Oct 16, 2014 #2
    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].

    Attached Files:

  4. Oct 16, 2014 #3
    You might want to check again.
  5. Oct 16, 2014 #4
    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.
    Last edited: Oct 16, 2014
  6. Oct 16, 2014 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    You should forget this bit of mathematical misinformation immediately.

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

    Here is a list of other derivatives:


    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
  7. Oct 16, 2014 #6


    User Avatar
    Homework Helper

    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.
  8. Oct 16, 2014 #7


    User Avatar
    Homework Helper

    Hint:arccos (x)=y means cos(y) = x.
  9. Oct 16, 2014 #8


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    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.
  10. Oct 17, 2014 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    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.
  11. Oct 19, 2014 #10
    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?
  12. Oct 20, 2014 #11


    User Avatar
    Homework Helper

    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.
  13. Oct 20, 2014 #12


    Staff: Mentor

    Let's hold off on any more replies until the OP comes back.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Proof of Arccos
  1. Arccos problems (Replies: 1)