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 hp49g+/hp48gII for ASIN was chosen because it is analytic in
the regions where the arguments of the real-valued inverse function are defined. The branch cut
for the complex-valued arc sine function occurs where the corresponding real-valued function is
undefined. The principal branch also preserves most of the important symmetries.
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 ASIN(Z)*(–1)^n1+ *n1 for the case
n1=0. For other values of n1, the vertical band in the lower graph is translated to the right (for n1
positive) or to the left (for n1 negative). Taken together, the bands cover the whole complex
plane, which is the domain of SIN.
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.
( ¼is the left-shift of the Skey).
Full Command and Function Reference 3-13