By definition, the inverse sine w = arc sinzis the relation such that sin w = z. The inverse cosine w €“ arcos z is the relation such that cos w = z. The inverse tangent, increase cotangent, inverse hyperbolic sine, etc, are defined and denoted in a similar fashion. (Note that all these relations are multi valued.) Using sin w = (e^iw€“ e^€“iw)/(2i) and similar representations of cos, w etc. show that 

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(a) arccos z = -i ln (z + Vz2 – 1) (b) arcsin z = -i In (iz + V1 - ") (c) arccosh z = In (z + Vz2 – 1) (d) arcsinh z = In (z + V2 + 1) In i- z (e) arctan z = In (f) arctanh z 1- z 2