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Is this equation correct

  1. Jan 8, 2013 #1

    utkarshakash

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    1. The problem statement, all variables and given/known data
    Is this equation correct for permissible values of x

    [itex]tan^{-1}|tan x| = |x|[/itex]

    2. Relevant equations

    3. The attempt at a solution
    I assume LHS to be θ.
    Then tanθ=|tanx|
    The original equation becomes
    [itex]tan^{-1}tan \theta = |x|[/itex]
     
  2. jcsd
  3. Jan 8, 2013 #2

    tiny-tim

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    hi utkarshakash! :smile:

    (what do you mean by "permissible"? :confused:)

    wouldn't it be easier to start by saying |tanx| = tan|x| ? :wink:
     
  4. Jan 8, 2013 #3

    HallsofIvy

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    No, that is not correct because tangent is not a "one-to-one function". For example, if [itex]\theta= 5\pi/4[/itex] then [itex]tan(\theta)= tan(5\pi/4)= 1[/itex] so that [itex]tan^{-1}(tan(\theta))= tan^{-1}(tan(5\pi/4))= tan^{-1}(1)= \pi/4[/itex], not [itex]5\pi/4[/itex]. Since everything is positive, the absolute value is irrelevant.
     
  5. Jan 8, 2013 #4
    Even if so, it is still wrong, as tangent can be negative for positive values of x. For instance, [itex]\tan(7\pi/4)=-1[/itex]. It is easy to observe from here that [itex]|\tan(7\pi/4)|\neq \tan|7\pi/4|[/itex], as [itex]1\neq-1[/itex].
     
  6. Jan 9, 2013 #5

    utkarshakash

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    Thanks all for your answers.
     
  7. Jan 9, 2013 #6

    haruspex

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    Another way to look at it is that atan(|y|) always produces a result in the range [0,pi/2). So the statement must be false for any |x| outside that range.
     
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