Results 1 to 3 of 3

Math Help - A question about composition of tigonometric functions

  1. #1
    Newbie
    Joined
    Nov 2011
    Posts
    1

    A question about composition of tigonometric functions

    A little something I'm trying to understand:
    sin(arcsin(x)) is always x, but
    arcsin(sin(x)) is not always x
    So my question is simple - why? Since each cancels the other, it would make sense that arcsin(sin(x)) would always result in x.
    I'd appreciate any explanation. Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,504
    Thanks
    1400

    Re: A question about composition of tigonometric functions

    Quote Originally Posted by yotamoo View Post
    A little something I'm trying to understand:
    So my question is simple - why? Since each cancels the other, it would make sense that arcsin(sin(x)) would always result in x.
    I'd appreciate any explanation. Thanks!
    Because adding any integer multiple of \displaystyle 2\pi also satisfies the solution to the second...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,710
    Thanks
    629

    Re: A question about composition of tigonometric functions

    Hello, yotamoo!

    A little something I'm trying to understand:

    . . \text{(a) }\sin(\arcsin x) is always x.

    . . \text{(b) }\arcsin(\sin x) is not always x.

    So my question is: why?

    I'll illustrate with a specific example.


    (a) We have: . \sin(\arcsin x)

    Suppose x = \tfrac{1}{2}

    Then: . \arcsin\left(\tfrac{1}{2}\right) \:=\:\begin{Bmatrix}\frac{\pi}{6} + 2\pi n \\ \\[-4mm] \frac{5\pi}{6} + 2\pi n \end{Bmatrix}

    Hence: . \sin\begin{pmatrix} \frac{\pi}{6} + 2\pi n \\ \frac{5\pi}{6} + 2\pi n \end{pmatrix} \:=\:\tfrac{1}{2}

    We get back our original value of x.



    (b) We have: . \arcsin(\sin x)

    Suppose x = \tfrac{5\pi}{6}

    Then: . \sin\tfrac{5\pi}{6} \,=\,\tfrac{1}{2}

    Hence: . \arcsin\left(\tfrac{1}{2}\right) \;=\;\begin{Bmatrix}\frac{\pi}{6} + 2\pi n \\ \frac{5\pi}{6} + 2\pi n \end{Bmatrix}

    There is an infinite number of possible values.
    We do not get back our original value of x.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: March 5th 2012, 04:34 PM
  2. Question about Composition Functions
    Posted in the Algebra Forum
    Replies: 12
    Last Post: September 13th 2011, 06:30 PM
  3. composition of functions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: August 29th 2010, 02:31 PM
  4. Another composition of functions question
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: July 17th 2010, 05:40 AM
  5. Replies: 1
    Last Post: November 5th 2009, 03:30 PM

Search Tags


/mathhelpforum @mathhelpforum