The inverse of SIN is a relation, not a function, since SIN sends more than one argument to the same result. The inverse relation for SIN is expressed by ISOL as the general solution:
The function ASIN is the inverse of a part of SIN, a part defined by restricting the domain of SIN such that:
•each argument is sent to a distinct result, and
•each possible result is achieved.
The points in this restricted domain of SIN are called the principal values of the inverse relation. ASIN in its entirety is called the principal branch of the inverse relation, and the points sent by ASIN to the boundary of the restricted domain of SIN form the branch cuts of ASIN.
The principal branch used by the calculator for ASIN was chosen because it is analytic in the regions where the arguments of the
The graphs below show the domain and range of ASIN. The graph of the domain shows where the branch cuts occur: the heavy solid line marks one side of a cut, while the feathered lines mark the other side of a cut. The graph of the range shows where each side of each cut is mapped under the function. These graphs show the inverse relation
View these graphs with domain and range reversed to see how the domain of SIN is restricted to make an inverse function possible. Consider the vertical band in the lower graph as the restricted domain Z = (x, y). SIN sends this domain onto the whole complex plane in the range
W = (u, v) = SIN(x, y) in the upper graph.
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Flags: Principal Solution