Results 1 to 5 of 5

Math Help - odd functions

  1. #1
    Junior Member
    Joined
    Oct 2006
    Posts
    71

    odd functions

    Show that arctan x is an odd function, that is, arctan –x = –arctan x.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by gracy View Post
    Show that arctan x is an odd function, that is, arctan –x = –arctan x.
    Let x=\tan(y), now \tan is an odd function so:

    -x=\tan(-y),

    so:

    <br />
\arctan(-x)=-y<br />

    but y=\arctan(x), hence:

    <br />
\arctan(-x)=-\arctan(x)<br />

    so \arctan is an odd function.

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by gracy View Post
    Show that arctan x is an odd function, that is, arctan –x = –arctan x.
    In general if f is invertible and odd.
    Then f^{-1} is invertible and odd.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,212
    Thanks
    419
    Awards
    1
    I'm not objecting to CaptainBlack's proof as much as I have a question about how to get around a problem with it. Given an angle x define y:
    y = tan(x)
    Then
    atn(y) = atn(tan(x)) \neq x in general because of the domain restriction we place on the atn function to make it bijective.

    I can easily see how restricting the domain of the tan function would fix this, but then we aren't really using the tan function. The only way I can think of to get around THIS is to extend the atn function so that it's no longer 1:1. But then it isn't really the inverse of the tan function any longer.

    I'm kinda going in circles here...

    Just wondering if it wouldn't be better to prove that atn is an odd function by using something more direct.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by topsquark View Post
    I'm not objecting to CaptainBlack's proof as much as I have a question about how to get around a problem with it. Given an angle x define y:
    y = tan(x)
    Then
    atn(y) = atn(tan(x)) \neq x in general because of the domain restriction we place on the atn function to make it bijective.

    I can easily see how restricting the domain of the tan function would fix this, but then we aren't really using the tan function. The only way I can think of to get around THIS is to extend the atn function so that it's no longer 1:1. But then it isn't really the inverse of the tan function any longer.

    I'm kinda going in circles here...

    Just wondering if it wouldn't be better to prove that atn is an odd function by using something more direct.

    -Dan
    There is an implicit restriction so that arctan takes value in the interval (pi/2,pi/2), and the domain of tan is also restricted to the same interval.

    (especialy as arctan is not odd on any other open interval of length pi).

    RonL
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: April 15th 2010, 06:50 PM
  2. Replies: 3
    Last Post: February 23rd 2010, 05:54 PM
  3. Replies: 11
    Last Post: November 15th 2009, 12:22 PM
  4. Replies: 7
    Last Post: August 12th 2009, 05:41 PM
  5. Replies: 1
    Last Post: April 15th 2008, 10:00 AM

Search Tags


/mathhelpforum @mathhelpforum