asin, acos, and atan
compute the arc sine, arc cosine, and arc tangent respectively.
The arc sine, arc cosine, and arc tangent (with only number1 supplied)
functions can be defined mathematically for
number or number1 specified as x as in the next figure.
Mathematical definition of arc sine, arc cosine, and arc tangent
Function |
Definition |
Arc sine |
-i log (ix+ sqrt(1-x^{2} )) |
Arc cosine |
( pi /2) - arcsin x |
Arc tangent |
-i log ((1+ix) sqrt(1/(1+x^{2}))) |
These formulae are mathematically correct, assuming
completely accurate computation. They are not necessarily
the simplest ones for real-valued computations.
If both number1 and number2 are supplied
for atan, the result is the arc tangent
of number1/number2.
The value of atan is always between
- pi (exclusive) and pi (inclusive)
when minus zero is not supported.
The range of the two-argument arc tangent when minus zero is supported
includes - pi .
For a
real
number1,
the result is
a real
and lies between
- pi /2 and pi /2 (both exclusive).
number1 can be a complex if number2
is not supplied. If both are supplied, number2 can be zero provided
number1 is not zero.
The following definition for arc sine determines the range and
branch cuts:
arcsin z = -i log (
iz+sqrt(1-z^{2}))
The branch cut for the arc sine function is in two pieces:
one along the negative real axis to the left of -1
(inclusive), continuous with quadrant II, and one along the positive real
axis to the right of 1 (inclusive), continuous with quadrant IV. The
range is that strip of the complex plane containing numbers whose real
part is between - pi /2 and pi /2. A number with real
part equal to - pi /2 is in the range if and only if its imaginary
part is non-negative; a number with real part equal to pi /2 is in
the range if and only if its imaginary part is non-positive.
The following definition for arc cosine determines the range and
branch cuts:
arccos z = pi /2- arcsin z
or, which are equivalent,
arccos z = -i log (z+i
sqrt(1-z^{2}))
arccos z = 2 log (sqrt((1+z)/2) +
i (sqrt(1-z)/2))/i
The branch cut for the arc cosine function is in two pieces:
one along the negative real axis to the left of -1
(inclusive), continuous with quadrant II, and one along the positive real
axis to the right of 1 (inclusive), continuous with quadrant IV.
This is the same branch cut as for arc sine.
The range is that strip of the complex plane containing numbers whose real
part is between 0 and pi . A number with real
part equal to 0 is in the range if and only if its imaginary
part is non-negative; a number with real part equal to pi is in
the range if and only if its imaginary part is non-positive.
The following definition for (one-argument) arc tangent determines the
range and branch cuts:
arctan z = log (1+iz) - log (1-iz)
/2i
Beware of simplifying this formula; "obvious" simplifications are likely
to alter the branch cuts or the values on the branch cuts incorrectly.
The branch cut for the arc tangent function is in two pieces:
one along the positive imaginary axis above i
(exclusive), continuous with quadrant II, and one along the negative imaginary
axis below -i (exclusive), continuous with quadrant IV.
The points i and -i are excluded from the domain.
The range is that strip of the complex plane containing numbers whose real
part is between - pi /2 and pi /2. A number with real
part equal to - pi /2 is in the range if and only if its imaginary
part is strictly positive; a number with real part equal to pi /2 is in
the range if and only if its imaginary part is strictly negative. Thus the range of
arc tangent is identical to that of arc sine with the points
- pi /2 and pi /2 excluded.
For atan,
the signs of number1 (indicated as x)
and number2 (indicated as y) are used to derive quadrant
information. The next figure details various special cases.
The asterisk (*) indicates that the entry in the figure applies to
implementations that support minus zero.
Quadrant information for arc tangent
y Condition |
x Condition |
Cartesian locus |
Range of result |
y = 0 |
x > 0 |
Positive x-axis |
0 |
* y = +0 |
x > 0 |
Positive x-axis |
+0 |
* y = -0 |
x > 0 |
Positive x-axis |
-0 |
y > 0 |
x > 0 |
Quadrant I |
0 < result < pi /2 |
y > 0 |
x = 0 |
Positive y-axis |
pi /2 |
y > 0 |
x < 0 |
Quadrant II |
pi /2 < result < pi |
y = 0 |
x < 0 |
Negative x-axis |
pi |
* y = +0 |
x < 0 |
Negative x-axis |
+ pi |
* y = -0 |
x < 0 |
Negative x-axis |
- pi |
y < 0 |
x < 0 |
Quadrant III |
- pi < result < - pi /2 |
y < 0 |
x = 0 |
Negative y-axis |
- pi /2 |
y < 0 |
x > 0 |
Quadrant IV |
- pi /2 < result < 0 |
y = 0 |
x = 0 |
Origin |
undefined consequences |
* y = +0 |
x = +0 |
Origin |
+0 |
* y = -0 |
x = +0 |
Origin |
-0 |
* y = +0 |
x = -0 |
Origin |
+ pi |
* y = -0 |
x = -0 |
Origin |
- pi |