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SIN(3M)								       SIN(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

       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  op‐
       erands and errno will be set to EDOM.

NOTES
       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	 frac‐
       tion  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), infnan(3M)

AUTHOR
       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)
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