sin, cos, tan, asin, acos, atan, atan2 - trigonometric functions and
double sin(double x)
double cos(double x)
double tan(double x)
double asin(double x)
double acos(double x)
double atan(double x)
double atan2(double y, double x)
Sin, cos and tan return trigonometric functions of radian arguments x.
Asin returns the arc sine in the range -pi/2 to pi/2.
Acos returns the arc cosine in the range 0 to pi.
Atan returns the arc tangent in the range -pi/2 to pi/2.
atan2(y, x) := atan(y/x) if x > 0,
sign(y)*(n - atan(|y/x|)) if x < 0,
0 if x = y = 0, or
sign(y)*n/2 if x = 0 != y.
sin(+Inf) = cos(+Inf) = tan(+Inf) = NaN with signal.
sin(NaN) = cos(NaN) = tan(NaN) = that NaN.
asin(NaN) = acos(NaN) = NaN.
asin(x) = acos(x) = NaN with invalid signal if |x| > 1.
atan2([anything], NaN) = NaN;
atan2(NaN , [anything]) = NaN;
atan2(+0, +[anything but NaN]) = +0 ;
atan2(+0, -[anything but NaN]) = +n ;
atan2(+[anything but 0 and NaN], 0) = +n/2;
atan2(+[anything but Inf and NaN], +Inf) = +0 ;
atan2(+[anything but Inf and NaN], -Inf) = +n;
atan2(+Inf, +Inf) = +n/4 ;
atan2(+Inf, -Inf) = +3n/4;
atan2(+Inf, [anything but, 0, NaN, and Inf]) = +n/2;
Atan2 defines atan2(0, 0) = 0 for the following reasons:
(1) Programs that test arguments to avoid computing atan2(0, 0) must
be indifferent to its value. Programs that require it to be
invalid are vulnerable to diverse reactions to that invalidity on
diverse computer systems.
(2) Atan2 is used mostly to convert from rectangular (x, y) to polar
(r, theta) coordinates that must satisfy x = r*cos theta and y =
r*sin theta. These equations are satisfied when (x=0, y=0) is
mapped to (r=0, theta=0) on a VAX. In general, conversions to
polar coordinates should be computed thus:
r := hypot(x, y); ... := sqrt(x*x+y*y)
theta := atan2(y, x).
(3) The foregoing formulas need not be altered to cope in a
reasonable way with signed zeros and infinities on a machine that
conforms to IEEE 754; the versions of hypot and atan2 provided
for such a machine are designed to handle all cases. That is why
atan2(+0, -0) = +pi, for instance. In general the formulas above
are equivalent to these:
r := sqrt(x*x+y*y); if r = 0 then x := copysign(1, x);
if x > 0 then theta := 2*atan(y/(r+x))
else theta := 2*atan((r-x)/y);
except if r is infinite then atan2 will yield an appropriate
multiple of pi/4 that would otherwise have to be obtained by
ERROR (due to Roundoff etc.)
Sin, cos, and tan return the nearly rounded result of the true
mathematical function, so they approximate within 1 ulp.
math(3), hypot(3), sqrt(3).
Robert P. Corbett, W. Kahan, Stuart I. McDonald, Peter Tang and, for the
codes for IEEE 754, Dr. Kwok-Choi Ng.