TRIG(3M)                                                 TRIG(3M)



     NAME
          sin, cos, tan, asin, acos, atan, atan2 - trigonometric
          functions 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
          If |x| > 1 then asin(x) and acos(x) returns NaN.

     NOTES
          Atan2 defines atan2(0,0) = 0.  The reasons for assigning a
          value to atan2(0,0) are these:

          (1) Programs that test arguments to avoid computing



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     TRIG(3M)                                                 TRIG(3M)



              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 engineering
          computations.

     SEE ALSO
          math(3M), hypot(3M), sqrt(3M)




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     TRIG(3M)                                                 TRIG(3M)



     AUTHOR
          Robert P. Corbett, W. Kahan, Stuart I. McDonald, Peter Tang
          and, for the codes for IEEE 754, Dr. Kwok-Choi Ng.

     ORIGIN
          MIPS Computer Systems

















































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