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Parity of inverse trigonometric functions

  1. Feb 16, 2014 #1
    When I place the trigonometric functions in the "wolfram google", it informs the parity of the function, so,

    sin(x), sinh(x) -> odd
    cos(x), cosh(x) -> even
    tan(x), tanh(x) -> odd
    cot(x), coth(x) -> odd
    sec(x), sech(x) -> even
    csc(x), csch(x) -> odd

    arcsin(x), arcsinh(x) -> odd
    arccos(x), arccosh(x) -> ???
    arctan(x), arctanh(x) -> odd
    arccot(x), arccoth(x) -> odd
    arcsec(x), arcsech(x) -> ???
    arccsc(x), arccsch(x) -> odd

    with base in this comparison above, is correct to attribute a parity for arccos(x), arccosh(x), arcsec(x) and arcsech(x) as being even?
  2. jcsd
  3. Feb 16, 2014 #2


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    In order to have an inverse, a function must be 1-1, hence it's inverse is also 1-1. In general, therefore, an inverse function cannot be even.

    To get the inverse of a trig function, the function is restricted to an interval where it is 1-1. For arcsin, this is [-π/2, π/2] and for cos [0, π] etc.
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