def sin(x, err=defaultError):
"""
sin(x [,err]) returns the sine of x.
"""
if not isinstance(err, defaultError.__class__) or err <= defaultError:
rx = rational.Rational(x)
sign = rational.Rational(1)
# sin(-x) = -sin(x)
if rx < 0:
sign = -sign
rx = -rx
# sin(x + 2 * pi) = sin(x)
if rx >= 2 * pi:
rx -= floor(rx / (pi * 2)) * (pi * 2)
# sin(x + pi) = -sin(x)
if rx >= pi:
rx -= pi
sign = -sign
# sin(x) = sin(pi - x)
if rx > pi / 2:
rx = pi - rx
# sin(0) = 0 is a special case which must not be computed with series.
if rx == 0:
return rational.Rational(0)
# sin(x) = cos(pi/2 - x) (pi/2 >= x > 4/pi)
if rx > pi / 4:
if rx == pi / 3:
retval = sqrt(3) / 2
else:
retval = _cosTaylor(pi / 2 - rx, err)
elif rx == pi / 4:
retval = 1 / sqrt(2)
elif rx == pi / 6:
retval = rational.Rational(1, 2)
else:
retval = _sinTaylor(rx, err)
if retval > 1:
retval = rational.Integer(1)
retval *= sign
else:
retval = rational.Rational(math.sin(x))
return retval
