# Acos - HP 48gII Advanced User's Reference Manual

Graphing calculator.

Input/Output: None
ACK

## ACOS

Type:
Analytic Function
Description: Arc Cosine Analytic Function: Returns the value of the angle having the given cosine.
For a real argument x in the domain –1 x 1, the result ranges from 0 to 180 degrees (0 to
A real argument outside of this domain is converted to a complex argument, z = x + 0i, and the
result is complex.
The inverse of COS is a relation, not a function, since COS sends more than one argument to the
same result. The inverse relation for COS is expressed by ISOL as the general solution
The function ACOS is the inverse of a part of COS, a part defined by restricting the domain of
COS such that:
• each argument is sent to a distinct result, and
• each possible result is achieved.
The points in this restricted domain of COS are called the principal values of the inverse relation.
ACOS in its entirety is called the principal branch of the inverse relation, and the points sent by
ACOS to the boundary of the restricted domain of COS form the branch cuts of ACOS.
The principal branch used by the hp49g+/hp48gII for ACOS 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 cosine 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 ACOS. 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 s1*ACOS(Z)+2* *n1 for the case s1=1 and n1 = 0. For
other values of s1 and n1, the vertical band in the lower graph is translated to the right or to the
left. Taken together, the bands cover the whole complex plane, which is the domain of COS.
View these graphs with domain and range reversed to see how the domain of COS is restricted
to make an inverse function possible. Consider the vertical band in the lower graph as the
restricted domain Z = (x, y). COS sends this domain onto the whole complex plane in the range
W = (u, v) = COS(x, y) in the upper graph.
Access:
Flags:
Principal Solution (–1), Numerical Results (–3), Angle Mode (–17, –18)
Input/Output:
ASIN, ATAN, COS, ISOL
s1*ACOS(Z)+2* *n1
( ¾ is the left-shift of the Tkey).
Level 1/Argument 1
z
'symb'
Full Command and Function Reference 3-5
Level 1/Item 1
acos z
'ACOS(symb)'