# 2.11BSD#431 - /usr/man/cat3/atan.0

    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 errno 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 11/26/99	  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 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 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 ulps 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 11/26/99	  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 11/26/99	  May 12, 1986				3

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