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There is no minimum value for the cosecant function.
An arccosecant is the function which is the compositional inverse of the cosecant function.
The cosecant is the reciprocal of the sine function. Now, the reciprocal of a positive number is positive, and the reciprocal of a negative number is negative.
It is an increasing odd function.
The inverse sine is the cosecant, otherwise known as "hypotenuse over opposite" or arcsine. The cosecant is often confused as being the inverse of the cosine, which, in reality, is the secant, otherwise known as "hypotenuse over adjacent" or arccosine.