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General inverses of trigonometry function

  1. Jun 20, 2010 #1
    I find this statement confusing from Wikipedia:

    [tex]sin(y)=x\Leftrightarrow y=arcsin(x)+2k\pi\ \forall\ k\in\mathbb Z[/tex]

    Is this statement false? "arcsin(x)" gives the principal value:[[itex]\frac{-\pi}{2},\frac{\pi}{2}[/itex]]. Therefore,specifically, "arcsin(x)" gives [itex]\frac{\pi}{3}\text{ but not}\frac{2\pi}{3}[/itex].

    My textbook:

    [tex]sin(y)=x\Leftrightarrow y=(-1)^{k}arcsin(x)+k\pi\ \forall\ k\in\mathbb Z[/tex]

    I think my textbook's statement is more "complete".Or are the 2 statements both true?
  2. jcsd
  3. Jun 20, 2010 #2


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    Homework Helper

    Yes your textbook is giving all the solutions, while wikipedia is missing about half of the possible solutions. Take [tex]y=2\pi/3[/tex] for example, wikipedia won't return that solution with its formula while the other will.
  4. Jun 20, 2010 #3


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    Look more closely at your text book. Most texts use "Arcsin(x)" to mean the "principal value" of arcsin(x)- the value y such that sin(y)= x for [itex]0\le y< \pi[/itex].

    Notice the difference? The CAPITALIZED A means the principal value, the small a arcsine, the more general value.
  5. Jun 20, 2010 #4
    Regarding capilatization conventions, Berkey/Blanchard: Calculus, 3rd ed. 1992, write, "Be careful to note that Tan-1 x means the inverse of the function y = Tan x , not (tan x)-1." Where Tan is the "restricted tangent function" with domain (-pi/2, pi/2) and range (-infinity, infinity). "The alternatives y = Arc tan x and y = arctan x are also frequently used to represent y = Tan-1 x." Even so, they call Tan-1, the "inverse tangent function", and Sin-1 the "inverse sine function" (for -1 <or= x <or= 1 and -pi/2 <or=y <or= pi/2, the inverse sine function is defined by y = Sin-1 x iff x = sin y). So for them, at least, it seems that arc- and Arc are interchangeable.
    Last edited: Jun 20, 2010
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