Digital

NAME

sin, cos, tan, fsin, fcos, ftan, asin, acos, atan, facos, fasin, fatan, atan2, fatan2 − trigonometric functions and their inverses

SYNOPSIS

#include <math.h>

double sin(x)
double x;

float fsin(float x)
float x;

double cos(x)
double x;

float fcos(float x)
float x;

double tan(float x)
double x;

float ftan(float x)
float x;

double asin(x)
double x;

float fasin(float x)
float x;

double acos(x)
double x;

float facos(float x)
float x;

double atan(x)
double x;

float fatan(float x)
float x;

double atan2(y,x)
double y,x;

float fatan2(float y,float x)
float y,x;

DESCRIPTION

sin, cos and tan return trigonometric functions of radian arguments x for double data types.  fsin, fcos and ftan do the same for float data types. 

asin and fasin returns the arc sine in the range −π/2 to π/2 for double and float data types respectively. 

acos and facos returns the arc cosine in the range 0 to π for double and float data types respectively. 

atan and fatan returns the arc tangent in the range −π/2 to π/2 for double and float data types respectively. 

atan2 and fatan2 returns the arctangent of y/x in the range −π to π, using the signs of both arguments to determine the quadrant of the return value for double and float data types respectively.

ERRORS

If |x| > 1 then asin(x) and acos(x) will return the default quiet 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 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,θ) coordinates that must satisfy x = r∗cosθ and y = r∗sinθ.  These equations are satisfied when (x=0,y=0) is mapped to (r=0,θ=0).  In general, conversions to polar coordinates should be computed thus:

r := hypot(x,y); ... := √(x2+y2)
θ := 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, such as MIPS machines, 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) = ±π, for instance.  In general the formulas above are equivalent to these:

r := √(x∗x+y∗y);  if r = 0 then x := copysign(1,x);
if x > 0thenθ := 2∗atan(y/(r+x))
elseθ := 2∗atan((r−x)/y);

except if r is infinite then atan2 will yield an appropriate multiple of π/4 that would otherwise have to be obtained by taking limits. 

ERROR (due to Roundoff etc.) for libm43.a only

Let P stand for the number stored in the computer in place of π = 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∗π/P), and "atrig(x)" approximates (P/π)∗atrig(x).  The approximations are close. 

In the codes that run on MIPS machines, P differs from π 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 π explicitly is satisfied equally well regardless of whether P = π.  For instance, sin2(x)+cos2(x) = 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 π is most unlikely to affect scientific and engineering computations. 

RELATED INFORMATION

hypot(3), math(3), sqrt(3). 

AUTHOR

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