4.3BSD-UWisc/man/cat3/sin.3m
SIN(3M) UNIX Programmer's Manual SIN(3M)
NAME
sin, cos, tan, asin, acos, atan, atan2 - trigonometric func-
tions and their inverses
SYNOPSIS
#include <math.h>
double sin(x)
double x;
double cos(x)
double x;
double tan(x)
double x;
double asin(x)
double x;
double acos(x)
double x;
double atan(x)
double x;
double atan2(y,x)
double y,x;
DESCRIPTION
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.
On a VAX,
atan2(y,x) := atan(y/x) if x > 0,
sign(y)*(pi - atan(|y/x|)) if x < 0,
0 if x = y = 0, or
sign(y)*pi/2 if x = 0 != y.
DIAGNOSTICS
On a VAX, if |x| > 1 then asin(x) and acos(x) will return
reserved operands and _e_r_r_n_o will be set to EDOM.
NOTES
Atan2 defines atan2(0,0) = 0 on a VAX despite that previ-
ously atan2(0,0) may have generated an error message. The
reasons for assigning a value to atan2(0,0) are these:
Printed 12/27/86 May 12, 1986 1
SIN(3M) UNIX Programmer's Manual SIN(3M)
(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 sys-
tems.
(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
taking limits.
ERROR (due to Roundoff etc.)
Let P stand for the number stored in the computer in place
of pi = 3.14159 26535 89793 23846 26433 ... . Let "trig"
stand for one of "sin", "cos" or "tan". Then the expression
"trig(x)" in a program actually produces an approximation to
trig(x*pi/P), and "atrig(x)" approximates (P/pi)*atrig(x).
The approximations are close, within 0.9 _u_l_ps for sin, cos
and atan, within 2.2 _u_l_ps for tan, asin, acos and atan2 on a
VAX. Moreover, P = pi in the codes that run on a VAX.
In the codes that run on other machines, P differs from pi
by a fraction of an _u_l_p; the difference matters only if the
argument x is huge, and even then the difference is likely
to be swamped by the uncertainty in x. Besides, every tri-
gonometric identity that does not involve pi explicitly is
satisfied equally well regardless of whether P = pi. For
instance, sin(x)**2+cos(x)**2 = 1 and
sin(2x) = 2sin(x)cos(x) to within a few _u_l_ps no matter how
big x may be. Therefore the difference between P and pi is
most unlikely to affect scientific and engineering computa-
tions.
Printed 12/27/86 May 12, 1986 2
SIN(3M) UNIX Programmer's Manual SIN(3M)
SEE ALSO
math(3M), hypot(3M), sqrt(3M), infnan(3M)
AUTHOR
Robert P. Corbett, W. Kahan, Stuart I. McDonald, Peter Tang
and, for the codes for IEEE 754, Dr. Kwok-Choi Ng.
Printed 12/27/86 May 12, 1986 3