The inverse cosecant function, in modern notation written as arccsc(x), gives the angle θ, so that:
Due to the periodical nature of the cosecant function, there are many angles θ that can give the same cosecant value (i.e. θ+2π, θ+4π, etc.). As a result, it is impossible to define a single inverse function, unless the range of the return values is restricted, so that a one-to-one relationship between θ and cscθ can be established. Therefore, multiple branches of the arccsc function can be defined. Commonly, the desired range of θ values spans between -π/2 and π/2. The branch of arccsc, in that case, is called the principal branch.
The arccsc function can be defined in a Taylor series form, like this:
The above series is valid for |x|≥1.
The derivative of the arccsc function is:
The integral of the arccsc function is given by:
The following properties are also valid for the arccsc function: