sin, cos, tan, asin, acos, atan, atan2 - trigonometric functions and their inverses
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
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.
On a VAX, if |x| > 1 then asin(x) and acos(x) will return reserved operands and errno will be set to EDOM.
Atan2 defines atan2(0,0) = 0 on a VAX despite that previously atan2(0,0) may have generated an error message. The reasons for assigning a
value to atan2(0,0) are these:(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 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 ulps for sin, cos and atan, within 2.2 ulps 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 ulp; 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 trigonometric 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 ulps no matter how big x may be. Therefore the difference between P and pi is most unlikely to affect scientific and engi-
math(3M), hypot(3M), sqrt(3M), infnan(3M)
Robert P. Corbett, W. Kahan, Stuart I. McDonald, Peter Tang and, for the codes for IEEE 754, Dr. Kwok-Choi Ng.
4th Berkeley Distribution May 12, 1986 SIN(3M)