from sympy.calculus.accumulationbounds import AccumBounds from sympy.core.add import Add from sympy.core.function import (Lambda, diff) from sympy.core.mul import Mul from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.complexes import (arg, conjugate, im, re) from sympy.functions.elementary.exponential import (exp, log) from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh, coth, sinh, tanh) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, atan, atan2, cos, cot, csc, sec, sin, sinc, tan) from sympy.functions.special.bessel import (besselj, jn) from sympy.functions.special.delta_functions import Heaviside from sympy.matrices.dense import Matrix from sympy.polys.polytools import (cancel, gcd) from sympy.series.limits import limit from sympy.series.order import O from sympy.series.series import series from sympy.sets.fancysets import ImageSet from sympy.sets.sets import (FiniteSet, Interval) from sympy.simplify.simplify import simplify from sympy.core.expr import unchanged from sympy.core.function import ArgumentIndexError from sympy.core.relational import Ne, Eq from sympy.functions.elementary.piecewise import Piecewise from sympy.sets.setexpr import SetExpr from sympy.testing.pytest import XFAIL, slow, raises x, y, z = symbols('x y z') r = Symbol('r', real=True) k, m = symbols('k m', integer=True) p = Symbol('p', positive=True) n = Symbol('n', negative=True) np = Symbol('p', nonpositive=True) nn = Symbol('n', nonnegative=True) nz = Symbol('nz', nonzero=True) ep = Symbol('ep', extended_positive=True) en = Symbol('en', extended_negative=True) enp = Symbol('ep', extended_nonpositive=True) enn = Symbol('en', extended_nonnegative=True) enz = Symbol('enz', extended_nonzero=True) a = Symbol('a', algebraic=True) na = Symbol('na', nonzero=True, algebraic=True) def test_sin(): x, y = symbols('x y') assert sin.nargs == FiniteSet(1) assert sin(nan) is nan assert sin(zoo) is nan assert sin(oo) == AccumBounds(-1, 1) assert sin(oo) - sin(oo) == AccumBounds(-2, 2) assert sin(oo*I) == oo*I assert sin(-oo*I) == -oo*I assert 0*sin(oo) is S.Zero assert 0/sin(oo) is S.Zero assert 0 + sin(oo) == AccumBounds(-1, 1) assert 5 + sin(oo) == AccumBounds(4, 6) assert sin(0) == 0 assert sin(asin(x)) == x assert sin(atan(x)) == x / sqrt(1 + x**2) assert sin(acos(x)) == sqrt(1 - x**2) assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x) assert sin(acsc(x)) == 1 / x assert sin(asec(x)) == sqrt(1 - 1 / x**2) assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2) assert sin(pi*I) == sinh(pi)*I assert sin(-pi*I) == -sinh(pi)*I assert sin(-2*I) == -sinh(2)*I assert sin(pi) == 0 assert sin(-pi) == 0 assert sin(2*pi) == 0 assert sin(-2*pi) == 0 assert sin(-3*10**73*pi) == 0 assert sin(7*10**103*pi) == 0 assert sin(pi/2) == 1 assert sin(-pi/2) == -1 assert sin(pi*Rational(5, 2)) == 1 assert sin(pi*Rational(7, 2)) == -1 ne = symbols('ne', integer=True, even=False) e = symbols('e', even=True) assert sin(pi*ne/2) == (-1)**(ne/2 - S.Half) assert sin(pi*k/2).func == sin assert sin(pi*e/2) == 0 assert sin(pi*k) == 0 assert sin(pi*k).subs(k, 3) == sin(pi*k/2).subs(k, 6) # issue 8298 assert sin(pi/3) == S.Half*sqrt(3) assert sin(pi*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3) assert sin(pi/4) == S.Half*sqrt(2) assert sin(-pi/4) == Rational(-1, 2)*sqrt(2) assert sin(pi*Rational(17, 4)) == S.Half*sqrt(2) assert sin(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2) assert sin(pi/6) == S.Half assert sin(-pi/6) == Rational(-1, 2) assert sin(pi*Rational(7, 6)) == Rational(-1, 2) assert sin(pi*Rational(-5, 6)) == Rational(-1, 2) assert sin(pi*Rational(1, 5)) == sqrt((5 - sqrt(5)) / 8) assert sin(pi*Rational(2, 5)) == sqrt((5 + sqrt(5)) / 8) assert sin(pi*Rational(3, 5)) == sin(pi*Rational(2, 5)) assert sin(pi*Rational(4, 5)) == sin(pi*Rational(1, 5)) assert sin(pi*Rational(6, 5)) == -sin(pi*Rational(1, 5)) assert sin(pi*Rational(8, 5)) == -sin(pi*Rational(2, 5)) assert sin(pi*Rational(-1273, 5)) == -sin(pi*Rational(2, 5)) assert sin(pi/8) == sqrt((2 - sqrt(2))/4) assert sin(pi/10) == Rational(-1, 4) + sqrt(5)/4 assert sin(pi/12) == -sqrt(2)/4 + sqrt(6)/4 assert sin(pi*Rational(5, 12)) == sqrt(2)/4 + sqrt(6)/4 assert sin(pi*Rational(-7, 12)) == -sqrt(2)/4 - sqrt(6)/4 assert sin(pi*Rational(-11, 12)) == sqrt(2)/4 - sqrt(6)/4 assert sin(pi*Rational(104, 105)) == sin(pi/105) assert sin(pi*Rational(106, 105)) == -sin(pi/105) assert sin(pi*Rational(-104, 105)) == -sin(pi/105) assert sin(pi*Rational(-106, 105)) == sin(pi/105) assert sin(x*I) == sinh(x)*I assert sin(k*pi) == 0 assert sin(17*k*pi) == 0 assert sin(2*k*pi + 4) == sin(4) assert sin(2*k*pi + m*pi + 1) == (-1)**(m + 2*k)*sin(1) assert sin(k*pi*I) == sinh(k*pi)*I assert sin(r).is_real is True assert sin(0, evaluate=False).is_algebraic assert sin(a).is_algebraic is None assert sin(na).is_algebraic is False q = Symbol('q', rational=True) assert sin(pi*q).is_algebraic qn = Symbol('qn', rational=True, nonzero=True) assert sin(qn).is_rational is False assert sin(q).is_rational is None # issue 8653 assert isinstance(sin( re(x) - im(y)), sin) is True assert isinstance(sin(-re(x) + im(y)), sin) is False assert sin(SetExpr(Interval(0, 1))) == SetExpr(ImageSet(Lambda(x, sin(x)), Interval(0, 1))) for d in list(range(1, 22)) + [60, 85]: for n in range(0, d*2 + 1): x = n*pi/d e = abs( float(sin(x)) - sin(float(x)) ) assert e < 1e-12 assert sin(0, evaluate=False).is_zero is True assert sin(k*pi, evaluate=False).is_zero is True assert sin(Add(1, -1, evaluate=False), evaluate=False).is_zero is True def test_sin_cos(): for d in [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 24, 30, 40, 60, 120]: # list is not exhaustive... for n in range(-2*d, d*2): x = n*pi/d assert sin(x + pi/2) == cos(x), "fails for %d*pi/%d" % (n, d) assert sin(x - pi/2) == -cos(x), "fails for %d*pi/%d" % (n, d) assert sin(x) == cos(x - pi/2), "fails for %d*pi/%d" % (n, d) assert -sin(x) == cos(x + pi/2), "fails for %d*pi/%d" % (n, d) def test_sin_series(): assert sin(x).series(x, 0, 9) == \ x - x**3/6 + x**5/120 - x**7/5040 + O(x**9) def test_sin_rewrite(): assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2 assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2) assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2) assert sin(sinh(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n() assert sin(cosh(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n() assert sin(tanh(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n() assert sin(coth(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n() assert sin(sin(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n() assert sin(cos(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n() assert sin(tan(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n() assert sin(cot(x)).rewrite( exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n() assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2 assert sin(x).rewrite(csc) == 1/csc(x) assert sin(x).rewrite(cos) == cos(x - pi / 2, evaluate=False) assert sin(x).rewrite(sec) == 1 / sec(x - pi / 2, evaluate=False) assert sin(cos(x)).rewrite(Pow) == sin(cos(x)) def _test_extrig(f, i, e): from sympy.core.function import expand_trig assert unchanged(f, i) assert expand_trig(f(i)) == f(i) # testing directly instead of with .expand(trig=True) # because the other expansions undo the unevaluated Mul assert expand_trig(f(Mul(i, 1, evaluate=False))) == e assert abs(f(i) - e).n() < 1e-10 def test_sin_expansion(): # Note: these formulas are not unique. The ones here come from the # Chebyshev formulas. assert sin(x + y).expand(trig=True) == sin(x)*cos(y) + cos(x)*sin(y) assert sin(x - y).expand(trig=True) == sin(x)*cos(y) - cos(x)*sin(y) assert sin(y - x).expand(trig=True) == cos(x)*sin(y) - sin(x)*cos(y) assert sin(2*x).expand(trig=True) == 2*sin(x)*cos(x) assert sin(3*x).expand(trig=True) == -4*sin(x)**3 + 3*sin(x) assert sin(4*x).expand(trig=True) == -8*sin(x)**3*cos(x) + 4*sin(x)*cos(x) _test_extrig(sin, 2, 2*sin(1)*cos(1)) _test_extrig(sin, 3, -4*sin(1)**3 + 3*sin(1)) def test_sin_AccumBounds(): assert sin(AccumBounds(-oo, oo)) == AccumBounds(-1, 1) assert sin(AccumBounds(0, oo)) == AccumBounds(-1, 1) assert sin(AccumBounds(-oo, 0)) == AccumBounds(-1, 1) assert sin(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1) assert sin(AccumBounds(0, S.Pi*Rational(3, 4))) == AccumBounds(0, 1) assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(7, 4))) == AccumBounds(-1, sin(S.Pi*Rational(3, 4))) assert sin(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(sin(S.Pi/4), sin(S.Pi/3)) assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 6))) == AccumBounds(sin(S.Pi*Rational(5, 6)), sin(S.Pi*Rational(3, 4))) def test_sin_fdiff(): assert sin(x).fdiff() == cos(x) raises(ArgumentIndexError, lambda: sin(x).fdiff(2)) def test_trig_symmetry(): assert sin(-x) == -sin(x) assert cos(-x) == cos(x) assert tan(-x) == -tan(x) assert cot(-x) == -cot(x) assert sin(x + pi) == -sin(x) assert sin(x + 2*pi) == sin(x) assert sin(x + 3*pi) == -sin(x) assert sin(x + 4*pi) == sin(x) assert sin(x - 5*pi) == -sin(x) assert cos(x + pi) == -cos(x) assert cos(x + 2*pi) == cos(x) assert cos(x + 3*pi) == -cos(x) assert cos(x + 4*pi) == cos(x) assert cos(x - 5*pi) == -cos(x) assert tan(x + pi) == tan(x) assert tan(x - 3*pi) == tan(x) assert cot(x + pi) == cot(x) assert cot(x - 3*pi) == cot(x) assert sin(pi/2 - x) == cos(x) assert sin(pi*Rational(3, 2) - x) == -cos(x) assert sin(pi*Rational(5, 2) - x) == cos(x) assert cos(pi/2 - x) == sin(x) assert cos(pi*Rational(3, 2) - x) == -sin(x) assert cos(pi*Rational(5, 2) - x) == sin(x) assert tan(pi/2 - x) == cot(x) assert tan(pi*Rational(3, 2) - x) == cot(x) assert tan(pi*Rational(5, 2) - x) == cot(x) assert cot(pi/2 - x) == tan(x) assert cot(pi*Rational(3, 2) - x) == tan(x) assert cot(pi*Rational(5, 2) - x) == tan(x) assert sin(pi/2 + x) == cos(x) assert cos(pi/2 + x) == -sin(x) assert tan(pi/2 + x) == -cot(x) assert cot(pi/2 + x) == -tan(x) def test_cos(): x, y = symbols('x y') assert cos.nargs == FiniteSet(1) assert cos(nan) is nan assert cos(oo) == AccumBounds(-1, 1) assert cos(oo) - cos(oo) == AccumBounds(-2, 2) assert cos(oo*I) is oo assert cos(-oo*I) is oo assert cos(zoo) is nan assert cos(0) == 1 assert cos(acos(x)) == x assert cos(atan(x)) == 1 / sqrt(1 + x**2) assert cos(asin(x)) == sqrt(1 - x**2) assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2) assert cos(acsc(x)) == sqrt(1 - 1 / x**2) assert cos(asec(x)) == 1 / x assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2) assert cos(pi*I) == cosh(pi) assert cos(-pi*I) == cosh(pi) assert cos(-2*I) == cosh(2) assert cos(pi/2) == 0 assert cos(-pi/2) == 0 assert cos(pi/2) == 0 assert cos(-pi/2) == 0 assert cos((-3*10**73 + 1)*pi/2) == 0 assert cos((7*10**103 + 1)*pi/2) == 0 n = symbols('n', integer=True, even=False) e = symbols('e', even=True) assert cos(pi*n/2) == 0 assert cos(pi*e/2) == (-1)**(e/2) assert cos(pi) == -1 assert cos(-pi) == -1 assert cos(2*pi) == 1 assert cos(5*pi) == -1 assert cos(8*pi) == 1 assert cos(pi/3) == S.Half assert cos(pi*Rational(-2, 3)) == Rational(-1, 2) assert cos(pi/4) == S.Half*sqrt(2) assert cos(-pi/4) == S.Half*sqrt(2) assert cos(pi*Rational(11, 4)) == Rational(-1, 2)*sqrt(2) assert cos(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2) assert cos(pi/6) == S.Half*sqrt(3) assert cos(-pi/6) == S.Half*sqrt(3) assert cos(pi*Rational(7, 6)) == Rational(-1, 2)*sqrt(3) assert cos(pi*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3) assert cos(pi*Rational(1, 5)) == (sqrt(5) + 1)/4 assert cos(pi*Rational(2, 5)) == (sqrt(5) - 1)/4 assert cos(pi*Rational(3, 5)) == -cos(pi*Rational(2, 5)) assert cos(pi*Rational(4, 5)) == -cos(pi*Rational(1, 5)) assert cos(pi*Rational(6, 5)) == -cos(pi*Rational(1, 5)) assert cos(pi*Rational(8, 5)) == cos(pi*Rational(2, 5)) assert cos(pi*Rational(-1273, 5)) == -cos(pi*Rational(2, 5)) assert cos(pi/8) == sqrt((2 + sqrt(2))/4) assert cos(pi/12) == sqrt(2)/4 + sqrt(6)/4 assert cos(pi*Rational(5, 12)) == -sqrt(2)/4 + sqrt(6)/4 assert cos(pi*Rational(7, 12)) == sqrt(2)/4 - sqrt(6)/4 assert cos(pi*Rational(11, 12)) == -sqrt(2)/4 - sqrt(6)/4 assert cos(pi*Rational(104, 105)) == -cos(pi/105) assert cos(pi*Rational(106, 105)) == -cos(pi/105) assert cos(pi*Rational(-104, 105)) == -cos(pi/105) assert cos(pi*Rational(-106, 105)) == -cos(pi/105) assert cos(x*I) == cosh(x) assert cos(k*pi*I) == cosh(k*pi) assert cos(r).is_real is True assert cos(0, evaluate=False).is_algebraic assert cos(a).is_algebraic is None assert cos(na).is_algebraic is False q = Symbol('q', rational=True) assert cos(pi*q).is_algebraic assert cos(pi*Rational(2, 7)).is_algebraic assert cos(k*pi) == (-1)**k assert cos(2*k*pi) == 1 for d in list(range(1, 22)) + [60, 85]: for n in range(0, 2*d + 1): x = n*pi/d e = abs( float(cos(x)) - cos(float(x)) ) assert e < 1e-12 def test_issue_6190(): c = Float('123456789012345678901234567890.25', '') for cls in [sin, cos, tan, cot]: assert cls(c*pi) == cls(pi/4) assert cls(4.125*pi) == cls(pi/8) assert cls(4.7*pi) == cls((4.7 % 2)*pi) def test_cos_series(): assert cos(x).series(x, 0, 9) == \ 1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9) def test_cos_rewrite(): assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2 assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2) assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2) assert cos(sinh(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n() assert cos(cosh(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n() assert cos(tanh(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n() assert cos(coth(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n() assert cos(sin(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n() assert cos(cos(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n() assert cos(tan(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n() assert cos(cot(x)).rewrite( exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n() assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2 assert cos(x).rewrite(sec) == 1/sec(x) assert cos(x).rewrite(sin) == sin(x + pi/2, evaluate=False) assert cos(x).rewrite(csc) == 1/csc(-x + pi/2, evaluate=False) assert cos(sin(x)).rewrite(Pow) == cos(sin(x)) def test_cos_expansion(): assert cos(x + y).expand(trig=True) == cos(x)*cos(y) - sin(x)*sin(y) assert cos(x - y).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y) assert cos(y - x).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y) assert cos(2*x).expand(trig=True) == 2*cos(x)**2 - 1 assert cos(3*x).expand(trig=True) == 4*cos(x)**3 - 3*cos(x) assert cos(4*x).expand(trig=True) == 8*cos(x)**4 - 8*cos(x)**2 + 1 _test_extrig(cos, 2, 2*cos(1)**2 - 1) _test_extrig(cos, 3, 4*cos(1)**3 - 3*cos(1)) def test_cos_AccumBounds(): assert cos(AccumBounds(-oo, oo)) == AccumBounds(-1, 1) assert cos(AccumBounds(0, oo)) == AccumBounds(-1, 1) assert cos(AccumBounds(-oo, 0)) == AccumBounds(-1, 1) assert cos(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1) assert cos(AccumBounds(-S.Pi/3, S.Pi/4)) == AccumBounds(cos(-S.Pi/3), 1) assert cos(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 4))) == AccumBounds(-1, cos(S.Pi*Rational(3, 4))) assert cos(AccumBounds(S.Pi*Rational(5, 4), S.Pi*Rational(4, 3))) == AccumBounds(cos(S.Pi*Rational(5, 4)), cos(S.Pi*Rational(4, 3))) assert cos(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(cos(S.Pi/3), cos(S.Pi/4)) def test_cos_fdiff(): assert cos(x).fdiff() == -sin(x) raises(ArgumentIndexError, lambda: cos(x).fdiff(2)) def test_tan(): assert tan(nan) is nan assert tan(zoo) is nan assert tan(oo) == AccumBounds(-oo, oo) assert tan(oo) - tan(oo) == AccumBounds(-oo, oo) assert tan.nargs == FiniteSet(1) assert tan(oo*I) == I assert tan(-oo*I) == -I assert tan(0) == 0 assert tan(atan(x)) == x assert tan(asin(x)) == x / sqrt(1 - x**2) assert tan(acos(x)) == sqrt(1 - x**2) / x assert tan(acot(x)) == 1 / x assert tan(acsc(x)) == 1 / (sqrt(1 - 1 / x**2) * x) assert tan(asec(x)) == sqrt(1 - 1 / x**2) * x assert tan(atan2(y, x)) == y/x assert tan(pi*I) == tanh(pi)*I assert tan(-pi*I) == -tanh(pi)*I assert tan(-2*I) == -tanh(2)*I assert tan(pi) == 0 assert tan(-pi) == 0 assert tan(2*pi) == 0 assert tan(-2*pi) == 0 assert tan(-3*10**73*pi) == 0 assert tan(pi/2) is zoo assert tan(pi*Rational(3, 2)) is zoo assert tan(pi/3) == sqrt(3) assert tan(pi*Rational(-2, 3)) == sqrt(3) assert tan(pi/4) is S.One assert tan(-pi/4) is S.NegativeOne assert tan(pi*Rational(17, 4)) is S.One assert tan(pi*Rational(-3, 4)) is S.One assert tan(pi/5) == sqrt(5 - 2*sqrt(5)) assert tan(pi*Rational(2, 5)) == sqrt(5 + 2*sqrt(5)) assert tan(pi*Rational(18, 5)) == -sqrt(5 + 2*sqrt(5)) assert tan(pi*Rational(-16, 5)) == -sqrt(5 - 2*sqrt(5)) assert tan(pi/6) == 1/sqrt(3) assert tan(-pi/6) == -1/sqrt(3) assert tan(pi*Rational(7, 6)) == 1/sqrt(3) assert tan(pi*Rational(-5, 6)) == 1/sqrt(3) assert tan(pi/8) == -1 + sqrt(2) assert tan(pi*Rational(3, 8)) == 1 + sqrt(2) # issue 15959 assert tan(pi*Rational(5, 8)) == -1 - sqrt(2) assert tan(pi*Rational(7, 8)) == 1 - sqrt(2) assert tan(pi/10) == sqrt(1 - 2*sqrt(5)/5) assert tan(pi*Rational(3, 10)) == sqrt(1 + 2*sqrt(5)/5) assert tan(pi*Rational(17, 10)) == -sqrt(1 + 2*sqrt(5)/5) assert tan(pi*Rational(-31, 10)) == -sqrt(1 - 2*sqrt(5)/5) assert tan(pi/12) == -sqrt(3) + 2 assert tan(pi*Rational(5, 12)) == sqrt(3) + 2 assert tan(pi*Rational(7, 12)) == -sqrt(3) - 2 assert tan(pi*Rational(11, 12)) == sqrt(3) - 2 assert tan(pi/24).radsimp() == -2 - sqrt(3) + sqrt(2) + sqrt(6) assert tan(pi*Rational(5, 24)).radsimp() == -2 + sqrt(3) - sqrt(2) + sqrt(6) assert tan(pi*Rational(7, 24)).radsimp() == 2 - sqrt(3) - sqrt(2) + sqrt(6) assert tan(pi*Rational(11, 24)).radsimp() == 2 + sqrt(3) + sqrt(2) + sqrt(6) assert tan(pi*Rational(13, 24)).radsimp() == -2 - sqrt(3) - sqrt(2) - sqrt(6) assert tan(pi*Rational(17, 24)).radsimp() == -2 + sqrt(3) + sqrt(2) - sqrt(6) assert tan(pi*Rational(19, 24)).radsimp() == 2 - sqrt(3) + sqrt(2) - sqrt(6) assert tan(pi*Rational(23, 24)).radsimp() == 2 + sqrt(3) - sqrt(2) - sqrt(6) assert tan(x*I) == tanh(x)*I assert tan(k*pi) == 0 assert tan(17*k*pi) == 0 assert tan(k*pi*I) == tanh(k*pi)*I assert tan(r).is_real is None assert tan(r).is_extended_real is True assert tan(0, evaluate=False).is_algebraic assert tan(a).is_algebraic is None assert tan(na).is_algebraic is False assert tan(pi*Rational(10, 7)) == tan(pi*Rational(3, 7)) assert tan(pi*Rational(11, 7)) == -tan(pi*Rational(3, 7)) assert tan(pi*Rational(-11, 7)) == tan(pi*Rational(3, 7)) assert tan(pi*Rational(15, 14)) == tan(pi/14) assert tan(pi*Rational(-15, 14)) == -tan(pi/14) assert tan(r).is_finite is None assert tan(I*r).is_finite is True # https://github.com/sympy/sympy/issues/21177 f = tan(pi*(x + S(3)/2))/(3*x) assert f.as_leading_term(x) == -1/(3*pi*x**2) def test_tan_series(): assert tan(x).series(x, 0, 9) == \ x + x**3/3 + 2*x**5/15 + 17*x**7/315 + O(x**9) def test_tan_rewrite(): neg_exp, pos_exp = exp(-x*I), exp(x*I) assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp) assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x) assert tan(x).rewrite(cos) == cos(x - S.Pi/2, evaluate=False)/cos(x) assert tan(x).rewrite(cot) == 1/cot(x) assert tan(sinh(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n() assert tan(cosh(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n() assert tan(tanh(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n() assert tan(coth(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n() assert tan(sin(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n() assert tan(cos(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n() assert tan(tan(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n() assert tan(cot(x)).rewrite( exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n() assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I) assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(pow) assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(pow) assert tan(pi/19).rewrite(pow) == tan(pi/19) assert tan(pi*Rational(8, 19)).rewrite(sqrt) == tan(pi*Rational(8, 19)) assert tan(x).rewrite(sec) == sec(x)/sec(x - pi/2, evaluate=False) assert tan(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)/csc(x) assert tan(sin(x)).rewrite(Pow) == tan(sin(x)) assert tan(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == sqrt(sqrt(5)/8 + Rational(5, 8))/(Rational(-1, 4) + sqrt(5)/4) def test_tan_subs(): assert tan(x).subs(tan(x), y) == y assert tan(x).subs(x, y) == tan(y) assert tan(x).subs(x, S.Pi/2) is zoo assert tan(x).subs(x, S.Pi*Rational(3, 2)) is zoo def test_tan_expansion(): assert tan(x + y).expand(trig=True) == ((tan(x) + tan(y))/(1 - tan(x)*tan(y))).expand() assert tan(x - y).expand(trig=True) == ((tan(x) - tan(y))/(1 + tan(x)*tan(y))).expand() assert tan(x + y + z).expand(trig=True) == ( (tan(x) + tan(y) + tan(z) - tan(x)*tan(y)*tan(z))/ (1 - tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z))).expand() assert 0 == tan(2*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 7))])*24 - 7 assert 0 == tan(3*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*55 - 37 assert 0 == tan(4*x - pi/4).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*239 - 1 _test_extrig(tan, 2, 2*tan(1)/(1 - tan(1)**2)) _test_extrig(tan, 3, (-tan(1)**3 + 3*tan(1))/(1 - 3*tan(1)**2)) def test_tan_AccumBounds(): assert tan(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo) assert tan(AccumBounds(S.Pi/3, S.Pi*Rational(2, 3))) == AccumBounds(-oo, oo) assert tan(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(tan(S.Pi/6), tan(S.Pi/3)) def test_tan_fdiff(): assert tan(x).fdiff() == tan(x)**2 + 1 raises(ArgumentIndexError, lambda: tan(x).fdiff(2)) def test_cot(): assert cot(nan) is nan assert cot.nargs == FiniteSet(1) assert cot(oo*I) == -I assert cot(-oo*I) == I assert cot(zoo) is nan assert cot(0) is zoo assert cot(2*pi) is zoo assert cot(acot(x)) == x assert cot(atan(x)) == 1 / x assert cot(asin(x)) == sqrt(1 - x**2) / x assert cot(acos(x)) == x / sqrt(1 - x**2) assert cot(acsc(x)) == sqrt(1 - 1 / x**2) * x assert cot(asec(x)) == 1 / (sqrt(1 - 1 / x**2) * x) assert cot(atan2(y, x)) == x/y assert cot(pi*I) == -coth(pi)*I assert cot(-pi*I) == coth(pi)*I assert cot(-2*I) == coth(2)*I assert cot(pi) == cot(2*pi) == cot(3*pi) assert cot(-pi) == cot(-2*pi) == cot(-3*pi) assert cot(pi/2) == 0 assert cot(-pi/2) == 0 assert cot(pi*Rational(5, 2)) == 0 assert cot(pi*Rational(7, 2)) == 0 assert cot(pi/3) == 1/sqrt(3) assert cot(pi*Rational(-2, 3)) == 1/sqrt(3) assert cot(pi/4) is S.One assert cot(-pi/4) is S.NegativeOne assert cot(pi*Rational(17, 4)) is S.One assert cot(pi*Rational(-3, 4)) is S.One assert cot(pi/6) == sqrt(3) assert cot(-pi/6) == -sqrt(3) assert cot(pi*Rational(7, 6)) == sqrt(3) assert cot(pi*Rational(-5, 6)) == sqrt(3) assert cot(pi/8) == 1 + sqrt(2) assert cot(pi*Rational(3, 8)) == -1 + sqrt(2) assert cot(pi*Rational(5, 8)) == 1 - sqrt(2) assert cot(pi*Rational(7, 8)) == -1 - sqrt(2) assert cot(pi/12) == sqrt(3) + 2 assert cot(pi*Rational(5, 12)) == -sqrt(3) + 2 assert cot(pi*Rational(7, 12)) == sqrt(3) - 2 assert cot(pi*Rational(11, 12)) == -sqrt(3) - 2 assert cot(pi/24).radsimp() == sqrt(2) + sqrt(3) + 2 + sqrt(6) assert cot(pi*Rational(5, 24)).radsimp() == -sqrt(2) - sqrt(3) + 2 + sqrt(6) assert cot(pi*Rational(7, 24)).radsimp() == -sqrt(2) + sqrt(3) - 2 + sqrt(6) assert cot(pi*Rational(11, 24)).radsimp() == sqrt(2) - sqrt(3) - 2 + sqrt(6) assert cot(pi*Rational(13, 24)).radsimp() == -sqrt(2) + sqrt(3) + 2 - sqrt(6) assert cot(pi*Rational(17, 24)).radsimp() == sqrt(2) - sqrt(3) + 2 - sqrt(6) assert cot(pi*Rational(19, 24)).radsimp() == sqrt(2) + sqrt(3) - 2 - sqrt(6) assert cot(pi*Rational(23, 24)).radsimp() == -sqrt(2) - sqrt(3) - 2 - sqrt(6) assert cot(x*I) == -coth(x)*I assert cot(k*pi*I) == -coth(k*pi)*I assert cot(r).is_real is None assert cot(r).is_extended_real is True assert cot(a).is_algebraic is None assert cot(na).is_algebraic is False assert cot(pi*Rational(10, 7)) == cot(pi*Rational(3, 7)) assert cot(pi*Rational(11, 7)) == -cot(pi*Rational(3, 7)) assert cot(pi*Rational(-11, 7)) == cot(pi*Rational(3, 7)) assert cot(pi*Rational(39, 34)) == cot(pi*Rational(5, 34)) assert cot(pi*Rational(-41, 34)) == -cot(pi*Rational(7, 34)) assert cot(x).is_finite is None assert cot(r).is_finite is None i = Symbol('i', imaginary=True) assert cot(i).is_finite is True assert cot(x).subs(x, 3*pi) is zoo # https://github.com/sympy/sympy/issues/21177 f = cot(pi*(x + 4))/(3*x) assert f.as_leading_term(x) == 1/(3*pi*x**2) def test_tan_cot_sin_cos_evalf(): assert abs((tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15)) - 1).evalf()) < 1e-14 assert abs((cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15)) - 1).evalf()) < 1e-14 @XFAIL def test_tan_cot_sin_cos_ratsimp(): assert 1 == (tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15))).ratsimp() assert 1 == (cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15))).ratsimp() def test_cot_series(): assert cot(x).series(x, 0, 9) == \ 1/x - x/3 - x**3/45 - 2*x**5/945 - x**7/4725 + O(x**9) # issue 6210 assert cot(x**4 + x**5).series(x, 0, 1) == \ x**(-4) - 1/x**3 + x**(-2) - 1/x + 1 + O(x) assert cot(pi*(1-x)).series(x, 0, 3) == -1/(pi*x) + pi*x/3 + O(x**3) assert cot(x).taylor_term(0, x) == 1/x assert cot(x).taylor_term(2, x) is S.Zero assert cot(x).taylor_term(3, x) == -x**3/45 def test_cot_rewrite(): neg_exp, pos_exp = exp(-x*I), exp(x*I) assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp) assert cot(x).rewrite(sin) == sin(2*x)/(2*(sin(x)**2)) assert cot(x).rewrite(cos) == cos(x)/cos(x - pi/2, evaluate=False) assert cot(x).rewrite(tan) == 1/tan(x) def check(func): z = cot(func(x)).rewrite(exp ) - cot(x).rewrite(exp).subs(x, func(x)) assert z.rewrite(exp).expand() == 0 check(sinh) check(cosh) check(tanh) check(coth) check(sin) check(cos) check(tan) assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I) assert cot(pi*Rational(4, 34)).rewrite(pow).ratsimp() == (cos(pi*Rational(4, 34))/sin(pi*Rational(4, 34))).rewrite(pow).ratsimp() assert cot(pi*Rational(4, 17)).rewrite(pow) == (cos(pi*Rational(4, 17))/sin(pi*Rational(4, 17))).rewrite(pow) assert cot(pi/19).rewrite(pow) == cot(pi/19) assert cot(pi/19).rewrite(sqrt) == cot(pi/19) assert cot(x).rewrite(sec) == sec(x - pi / 2, evaluate=False) / sec(x) assert cot(x).rewrite(csc) == csc(x) / csc(- x + pi / 2, evaluate=False) assert cot(sin(x)).rewrite(Pow) == cot(sin(x)) assert cot(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == (Rational(-1, 4) + sqrt(5)/4)/\ sqrt(sqrt(5)/8 + Rational(5, 8)) def test_cot_subs(): assert cot(x).subs(cot(x), y) == y assert cot(x).subs(x, y) == cot(y) assert cot(x).subs(x, 0) is zoo assert cot(x).subs(x, S.Pi) is zoo def test_cot_expansion(): assert cot(x + y).expand(trig=True).together() == ( (cot(x)*cot(y) - 1)/(cot(x) + cot(y))) assert cot(x - y).expand(trig=True).together() == ( cot(x)*cot(-y) - 1)/(cot(x) + cot(-y)) assert cot(x + y + z).expand(trig=True).together() == ( (cot(x)*cot(y)*cot(z) - cot(x) - cot(y) - cot(z))/ (-1 + cot(x)*cot(y) + cot(x)*cot(z) + cot(y)*cot(z))) assert cot(3*x).expand(trig=True).together() == ( (cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1)) assert cot(2*x).expand(trig=True) == cot(x)/2 - 1/(2*cot(x)) assert cot(3*x).expand(trig=True).together() == ( cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1) assert cot(4*x - pi/4).expand(trig=True).cancel() == ( -tan(x)**4 + 4*tan(x)**3 + 6*tan(x)**2 - 4*tan(x) - 1 )/(tan(x)**4 + 4*tan(x)**3 - 6*tan(x)**2 - 4*tan(x) + 1) _test_extrig(cot, 2, (-1 + cot(1)**2)/(2*cot(1))) _test_extrig(cot, 3, (-3*cot(1) + cot(1)**3)/(-1 + 3*cot(1)**2)) def test_cot_AccumBounds(): assert cot(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo) assert cot(AccumBounds(-S.Pi/3, S.Pi/3)) == AccumBounds(-oo, oo) assert cot(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(cot(S.Pi/3), cot(S.Pi/6)) def test_cot_fdiff(): assert cot(x).fdiff() == -cot(x)**2 - 1 raises(ArgumentIndexError, lambda: cot(x).fdiff(2)) def test_sinc(): assert isinstance(sinc(x), sinc) s = Symbol('s', zero=True) assert sinc(s) is S.One assert sinc(S.Infinity) is S.Zero assert sinc(S.NegativeInfinity) is S.Zero assert sinc(S.NaN) is S.NaN assert sinc(S.ComplexInfinity) is S.NaN n = Symbol('n', integer=True, nonzero=True) assert sinc(n*pi) is S.Zero assert sinc(-n*pi) is S.Zero assert sinc(pi/2) == 2 / pi assert sinc(-pi/2) == 2 / pi assert sinc(pi*Rational(5, 2)) == 2 / (5*pi) assert sinc(pi*Rational(7, 2)) == -2 / (7*pi) assert sinc(-x) == sinc(x) assert sinc(x).diff(x) == cos(x)/x - sin(x)/x**2 assert sinc(x).diff(x) == (sin(x)/x).diff(x) assert sinc(x).diff(x, x) == (-sin(x) - 2*cos(x)/x + 2*sin(x)/x**2)/x assert sinc(x).diff(x, x) == (sin(x)/x).diff(x, x) assert limit(sinc(x).diff(x), x, 0) == 0 assert limit(sinc(x).diff(x, x), x, 0) == -S(1)/3 # https://github.com/sympy/sympy/issues/11402 # # assert sinc(x).diff(x) == Piecewise(((x*cos(x) - sin(x)) / x**2, Ne(x, 0)), (0, True)) # # assert sinc(x).diff(x).equals(sinc(x).rewrite(sin).diff(x)) # # assert sinc(x).diff(x).subs(x, 0) is S.Zero assert sinc(x).series() == 1 - x**2/6 + x**4/120 + O(x**6) assert sinc(x).rewrite(jn) == jn(0, x) assert sinc(x).rewrite(sin) == Piecewise((sin(x)/x, Ne(x, 0)), (1, True)) assert sinc(pi, evaluate=False).is_zero is True assert sinc(0, evaluate=False).is_zero is False assert sinc(n*pi, evaluate=False).is_zero is True assert sinc(x).is_zero is None xr = Symbol('xr', real=True, nonzero=True) assert sinc(x).is_real is None assert sinc(xr).is_real is True assert sinc(I*xr).is_real is True assert sinc(I*100).is_real is True assert sinc(x).is_finite is None assert sinc(xr).is_finite is True def test_asin(): assert asin(nan) is nan assert asin.nargs == FiniteSet(1) assert asin(oo) == -I*oo assert asin(-oo) == I*oo assert asin(zoo) is zoo # Note: asin(-x) = - asin(x) assert asin(0) == 0 assert asin(1) == pi/2 assert asin(-1) == -pi/2 assert asin(sqrt(3)/2) == pi/3 assert asin(-sqrt(3)/2) == -pi/3 assert asin(sqrt(2)/2) == pi/4 assert asin(-sqrt(2)/2) == -pi/4 assert asin(sqrt((5 - sqrt(5))/8)) == pi/5 assert asin(-sqrt((5 - sqrt(5))/8)) == -pi/5 assert asin(S.Half) == pi/6 assert asin(Rational(-1, 2)) == -pi/6 assert asin((sqrt(2 - sqrt(2)))/2) == pi/8 assert asin(-(sqrt(2 - sqrt(2)))/2) == -pi/8 assert asin((sqrt(5) - 1)/4) == pi/10 assert asin(-(sqrt(5) - 1)/4) == -pi/10 assert asin((sqrt(3) - 1)/sqrt(2**3)) == pi/12 assert asin(-(sqrt(3) - 1)/sqrt(2**3)) == -pi/12 # check round-trip for exact values: for d in [5, 6, 8, 10, 12]: for n in range(-(d//2), d//2 + 1): if gcd(n, d) == 1: assert asin(sin(n*pi/d)) == n*pi/d assert asin(x).diff(x) == 1/sqrt(1 - x**2) assert asin(1/x).as_leading_term(x) == I*log(1/x) assert asin(0.2, evaluate=False).is_real is True assert asin(-2).is_real is False assert asin(r).is_real is None assert asin(-2*I) == -I*asinh(2) assert asin(Rational(1, 7), evaluate=False).is_positive is True assert asin(Rational(-1, 7), evaluate=False).is_positive is False assert asin(p).is_positive is None assert asin(sin(Rational(7, 2))) == Rational(-7, 2) + pi assert asin(sin(Rational(-7, 4))) == Rational(7, 4) - pi assert unchanged(asin, cos(x)) def test_asin_series(): assert asin(x).series(x, 0, 9) == \ x + x**3/6 + 3*x**5/40 + 5*x**7/112 + O(x**9) t5 = asin(x).taylor_term(5, x) assert t5 == 3*x**5/40 assert asin(x).taylor_term(7, x, t5, 0) == 5*x**7/112 def test_asin_rewrite(): assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2)) assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2))) assert asin(x).rewrite(acos) == S.Pi/2 - acos(x) assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x) assert asin(x).rewrite(asec) == -asec(1/x) + pi/2 assert asin(x).rewrite(acsc) == acsc(1/x) def test_asin_fdiff(): assert asin(x).fdiff() == 1/sqrt(1 - x**2) raises(ArgumentIndexError, lambda: asin(x).fdiff(2)) def test_acos(): assert acos(nan) is nan assert acos(zoo) is zoo assert acos.nargs == FiniteSet(1) assert acos(oo) == I*oo assert acos(-oo) == -I*oo # Note: acos(-x) = pi - acos(x) assert acos(0) == pi/2 assert acos(S.Half) == pi/3 assert acos(Rational(-1, 2)) == pi*Rational(2, 3) assert acos(1) == 0 assert acos(-1) == pi assert acos(sqrt(2)/2) == pi/4 assert acos(-sqrt(2)/2) == pi*Rational(3, 4) # check round-trip for exact values: for d in [5, 6, 8, 10, 12]: for num in range(d): if gcd(num, d) == 1: assert acos(cos(num*pi/d)) == num*pi/d assert acos(2*I) == pi/2 - asin(2*I) assert acos(x).diff(x) == -1/sqrt(1 - x**2) assert acos(1/x).as_leading_term(x) == I*log(1/x) assert acos(0.2).is_real is True assert acos(-2).is_real is False assert acos(r).is_real is None assert acos(Rational(1, 7), evaluate=False).is_positive is True assert acos(Rational(-1, 7), evaluate=False).is_positive is True assert acos(Rational(3, 2), evaluate=False).is_positive is False assert acos(p).is_positive is None assert acos(2 + p).conjugate() != acos(10 + p) assert acos(-3 + n).conjugate() != acos(-3 + n) assert acos(Rational(1, 3)).conjugate() == acos(Rational(1, 3)) assert acos(Rational(-1, 3)).conjugate() == acos(Rational(-1, 3)) assert acos(p + n*I).conjugate() == acos(p - n*I) assert acos(z).conjugate() != acos(conjugate(z)) def test_acos_series(): assert acos(x).series(x, 0, 8) == \ pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8) assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8) t5 = acos(x).taylor_term(5, x) assert t5 == -3*x**5/40 assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112 assert acos(x).taylor_term(0, x) == pi/2 assert acos(x).taylor_term(2, x) is S.Zero def test_acos_rewrite(): assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2)) assert acos(x).rewrite(atan) == \ atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2)) assert acos(0).rewrite(atan) == S.Pi/2 assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log) assert acos(x).rewrite(asin) == S.Pi/2 - asin(x) assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2 assert acos(x).rewrite(asec) == asec(1/x) assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2 def test_acos_fdiff(): assert acos(x).fdiff() == -1/sqrt(1 - x**2) raises(ArgumentIndexError, lambda: acos(x).fdiff(2)) def test_atan(): assert atan(nan) is nan assert atan.nargs == FiniteSet(1) assert atan(oo) == pi/2 assert atan(-oo) == -pi/2 assert atan(zoo) == AccumBounds(-pi/2, pi/2) assert atan(0) == 0 assert atan(1) == pi/4 assert atan(sqrt(3)) == pi/3 assert atan(-(1 + sqrt(2))) == pi*Rational(-3, 8) assert atan(sqrt(5 - 2 * sqrt(5))) == pi/5 assert atan(-sqrt(1 - 2 * sqrt(5)/ 5)) == -pi/10 assert atan(sqrt(1 + 2 * sqrt(5) / 5)) == pi*Rational(3, 10) assert atan(-2 + sqrt(3)) == -pi/12 assert atan(2 + sqrt(3)) == pi*Rational(5, 12) assert atan(-2 - sqrt(3)) == pi*Rational(-5, 12) # check round-trip for exact values: for d in [5, 6, 8, 10, 12]: for num in range(-(d//2), d//2 + 1): if gcd(num, d) == 1: assert atan(tan(num*pi/d)) == num*pi/d assert atan(oo) == pi/2 assert atan(x).diff(x) == 1/(1 + x**2) assert atan(1/x).as_leading_term(x) == pi/2 assert atan(r).is_real is True assert atan(-2*I) == -I*atanh(2) assert unchanged(atan, cot(x)) assert atan(cot(Rational(1, 4))) == Rational(-1, 4) + pi/2 assert acot(Rational(1, 4)).is_rational is False for s in (x, p, n, np, nn, nz, ep, en, enp, enn, enz): if s.is_real or s.is_extended_real is None: assert s.is_nonzero is atan(s).is_nonzero assert s.is_positive is atan(s).is_positive assert s.is_negative is atan(s).is_negative assert s.is_nonpositive is atan(s).is_nonpositive assert s.is_nonnegative is atan(s).is_nonnegative else: assert s.is_extended_nonzero is atan(s).is_nonzero assert s.is_extended_positive is atan(s).is_positive assert s.is_extended_negative is atan(s).is_negative assert s.is_extended_nonpositive is atan(s).is_nonpositive assert s.is_extended_nonnegative is atan(s).is_nonnegative assert s.is_extended_nonzero is atan(s).is_extended_nonzero assert s.is_extended_positive is atan(s).is_extended_positive assert s.is_extended_negative is atan(s).is_extended_negative assert s.is_extended_nonpositive is atan(s).is_extended_nonpositive assert s.is_extended_nonnegative is atan(s).is_extended_nonnegative def test_atan_rewrite(): assert atan(x).rewrite(log) == I*(log(1 - I*x)-log(1 + I*x))/2 assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x assert atan(x).rewrite(acot) == acot(1/x) assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x assert atan(-5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:-5*I}) assert atan(5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:5*I}) def test_atan_fdiff(): assert atan(x).fdiff() == 1/(x**2 + 1) raises(ArgumentIndexError, lambda: atan(x).fdiff(2)) def test_atan2(): assert atan2.nargs == FiniteSet(2) assert atan2(0, 0) is S.NaN assert atan2(0, 1) == 0 assert atan2(1, 1) == pi/4 assert atan2(1, 0) == pi/2 assert atan2(1, -1) == pi*Rational(3, 4) assert atan2(0, -1) == pi assert atan2(-1, -1) == pi*Rational(-3, 4) assert atan2(-1, 0) == -pi/2 assert atan2(-1, 1) == -pi/4 i = symbols('i', imaginary=True) r = symbols('r', real=True) eq = atan2(r, i) ans = -I*log((i + I*r)/sqrt(i**2 + r**2)) reps = ((r, 2), (i, I)) assert eq.subs(reps) == ans.subs(reps) x = Symbol('x', negative=True) y = Symbol('y', negative=True) assert atan2(y, x) == atan(y/x) - pi y = Symbol('y', nonnegative=True) assert atan2(y, x) == atan(y/x) + pi y = Symbol('y') assert atan2(y, x) == atan2(y, x, evaluate=False) u = Symbol("u", positive=True) assert atan2(0, u) == 0 u = Symbol("u", negative=True) assert atan2(0, u) == pi assert atan2(y, oo) == 0 assert atan2(y, -oo)== 2*pi*Heaviside(re(y), S.Half) - pi assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2)) assert atan2(0, 0) is S.NaN ex = atan2(y, x) - arg(x + I*y) assert ex.subs({x:2, y:3}).rewrite(arg) == 0 assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5) assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I) assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(Rational(2, 3)) + atan(Rational(3, 2)) i = symbols('i', imaginary=True) r = symbols('r', real=True) e = atan2(i, r) rewrite = e.rewrite(arg) reps = {i: I, r: -2} assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2)) assert (e - rewrite).subs(reps).equals(0) assert atan2(0, x).rewrite(atan) == Piecewise((pi, re(x) < 0), (0, Ne(x, 0)), (nan, True)) assert atan2(0, r).rewrite(atan) == Piecewise((pi, r < 0), (0, Ne(r, 0)), (S.NaN, True)) assert atan2(0, i),rewrite(atan) == 0 assert atan2(0, r + i).rewrite(atan) == Piecewise((pi, r < 0), (0, True)) assert atan2(y, x).rewrite(atan) == Piecewise( (2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, (re(x) > 0) | Ne(im(x), 0)), (nan, True)) assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y)) assert diff(atan2(y, x), x) == -y/(x**2 + y**2) assert diff(atan2(y, x), y) == x/(x**2 + y**2) assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2) assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2) assert str(atan2(1, 2).evalf(5)) == '0.46365' raises(ArgumentIndexError, lambda: atan2(x, y).fdiff(3)) def test_issue_17461(): class A(Symbol): is_extended_real = True def _eval_evalf(self, prec): return Float(5.0) x = A('X') y = A('Y') assert abs(atan2(x, y).evalf() - 0.785398163397448) <= 1e-10 def test_acot(): assert acot(nan) is nan assert acot.nargs == FiniteSet(1) assert acot(-oo) == 0 assert acot(oo) == 0 assert acot(zoo) == 0 assert acot(1) == pi/4 assert acot(0) == pi/2 assert acot(sqrt(3)/3) == pi/3 assert acot(1/sqrt(3)) == pi/3 assert acot(-1/sqrt(3)) == -pi/3 assert acot(x).diff(x) == -1/(1 + x**2) assert acot(1/x).as_leading_term(x) == x assert acot(r).is_extended_real is True assert acot(I*pi) == -I*acoth(pi) assert acot(-2*I) == I*acoth(2) assert acot(x).is_positive is None assert acot(n).is_positive is False assert acot(p).is_positive is True assert acot(I).is_positive is False assert acot(Rational(1, 4)).is_rational is False assert unchanged(acot, cot(x)) assert unchanged(acot, tan(x)) assert acot(cot(Rational(1, 4))) == Rational(1, 4) assert acot(tan(Rational(-1, 4))) == Rational(1, 4) - pi/2 def test_acot_rewrite(): assert acot(x).rewrite(log) == I*(log(1 - I/x)-log(1 + I/x))/2 assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2)) assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1)) assert acot(x).rewrite(atan) == atan(1/x) assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2)) assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2)) assert acot(-I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:-I/5}) assert acot(I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:I/5}) def test_acot_fdiff(): assert acot(x).fdiff() == -1/(x**2 + 1) raises(ArgumentIndexError, lambda: acot(x).fdiff(2)) def test_attributes(): assert sin(x).args == (x,) def test_sincos_rewrite(): assert sin(pi/2 - x) == cos(x) assert sin(pi - x) == sin(x) assert cos(pi/2 - x) == sin(x) assert cos(pi - x) == -cos(x) def _check_even_rewrite(func, arg): """Checks that the expr has been rewritten using f(-x) -> f(x) arg : -x """ return func(arg).args[0] == -arg def _check_odd_rewrite(func, arg): """Checks that the expr has been rewritten using f(-x) -> -f(x) arg : -x """ return func(arg).func.is_Mul def _check_no_rewrite(func, arg): """Checks that the expr is not rewritten""" return func(arg).args[0] == arg def test_evenodd_rewrite(): a = cos(2) # negative b = sin(1) # positive even = [cos] odd = [sin, tan, cot, asin, atan, acot] with_minus = [-1, -2**1024 * E, -pi/105, -x*y, -x - y] for func in even: for expr in with_minus: assert _check_even_rewrite(func, expr) assert _check_no_rewrite(func, a*b) assert func( x - y) == func(y - x) # it doesn't matter which form is canonical for func in odd: for expr in with_minus: assert _check_odd_rewrite(func, expr) assert _check_no_rewrite(func, a*b) assert func( x - y) == -func(y - x) # it doesn't matter which form is canonical def test_issue_4547(): assert sin(x).rewrite(cot) == 2*cot(x/2)/(1 + cot(x/2)**2) assert cos(x).rewrite(cot) == -(1 - cot(x/2)**2)/(1 + cot(x/2)**2) assert tan(x).rewrite(cot) == 1/cot(x) assert cot(x).fdiff() == -1 - cot(x)**2 def test_as_leading_term_issue_5272(): assert sin(x).as_leading_term(x) == x assert cos(x).as_leading_term(x) == 1 assert tan(x).as_leading_term(x) == x assert cot(x).as_leading_term(x) == 1/x assert asin(x).as_leading_term(x) == x assert acos(x).as_leading_term(x) == pi/2 assert atan(x).as_leading_term(x) == x assert acot(x).as_leading_term(x) == pi/2 def test_leading_terms(): assert sin(1/x).as_leading_term(x) == AccumBounds(-1, 1) assert sin(S.Half).as_leading_term(x) == sin(S.Half) assert cos(1/x).as_leading_term(x) == AccumBounds(-1, 1) assert cos(S.Half).as_leading_term(x) == cos(S.Half) for func in [tan, cot]: for a in (1/x, S.Half): eq = func(a) assert eq.as_leading_term(x) == eq # https://github.com/sympy/sympy/issues/21038 f = sin(pi*(x + 4))/(3*x) assert f.as_leading_term(x) == pi/3 def test_atan2_expansion(): assert cancel(atan2(x**2, x + 1).diff(x) - atan(x**2/(x + 1)).diff(x)) == 0 assert cancel(atan(y/x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5) + atan2(0, x) - atan(0)) == O(y**5) assert cancel(atan(y/x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4) + atan2(y, 1) - atan(y)) == O((x - 1)**4, (x, 1)) assert cancel(atan((y + x)/x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3) + atan2(1 + y, 1) - atan(1 + y)) == O((x - 1)**3, (x, 1)) assert Matrix([atan2(y, x)]).jacobian([y, x]) == \ Matrix([[x/(y**2 + x**2), -y/(y**2 + x**2)]]) def test_aseries(): def t(n, v, d, e): assert abs( n(1/v).evalf() - n(1/x).series(x, dir=d).removeO().subs(x, v)) < e t(atan, 0.1, '+', 1e-5) t(atan, -0.1, '-', 1e-5) t(acot, 0.1, '+', 1e-5) t(acot, -0.1, '-', 1e-5) def test_issue_4420(): i = Symbol('i', integer=True) e = Symbol('e', even=True) o = Symbol('o', odd=True) # unknown parity for variable assert cos(4*i*pi) == 1 assert sin(4*i*pi) == 0 assert tan(4*i*pi) == 0 assert cot(4*i*pi) is zoo assert cos(3*i*pi) == cos(pi*i) # +/-1 assert sin(3*i*pi) == 0 assert tan(3*i*pi) == 0 assert cot(3*i*pi) is zoo assert cos(4.0*i*pi) == 1 assert sin(4.0*i*pi) == 0 assert tan(4.0*i*pi) == 0 assert cot(4.0*i*pi) is zoo assert cos(3.0*i*pi) == cos(pi*i) # +/-1 assert sin(3.0*i*pi) == 0 assert tan(3.0*i*pi) == 0 assert cot(3.0*i*pi) is zoo assert cos(4.5*i*pi) == cos(0.5*pi*i) assert sin(4.5*i*pi) == sin(0.5*pi*i) assert tan(4.5*i*pi) == tan(0.5*pi*i) assert cot(4.5*i*pi) == cot(0.5*pi*i) # parity of variable is known assert cos(4*e*pi) == 1 assert sin(4*e*pi) == 0 assert tan(4*e*pi) == 0 assert cot(4*e*pi) is zoo assert cos(3*e*pi) == 1 assert sin(3*e*pi) == 0 assert tan(3*e*pi) == 0 assert cot(3*e*pi) is zoo assert cos(4.0*e*pi) == 1 assert sin(4.0*e*pi) == 0 assert tan(4.0*e*pi) == 0 assert cot(4.0*e*pi) is zoo assert cos(3.0*e*pi) == 1 assert sin(3.0*e*pi) == 0 assert tan(3.0*e*pi) == 0 assert cot(3.0*e*pi) is zoo assert cos(4.5*e*pi) == cos(0.5*pi*e) assert sin(4.5*e*pi) == sin(0.5*pi*e) assert tan(4.5*e*pi) == tan(0.5*pi*e) assert cot(4.5*e*pi) == cot(0.5*pi*e) assert cos(4*o*pi) == 1 assert sin(4*o*pi) == 0 assert tan(4*o*pi) == 0 assert cot(4*o*pi) is zoo assert cos(3*o*pi) == -1 assert sin(3*o*pi) == 0 assert tan(3*o*pi) == 0 assert cot(3*o*pi) is zoo assert cos(4.0*o*pi) == 1 assert sin(4.0*o*pi) == 0 assert tan(4.0*o*pi) == 0 assert cot(4.0*o*pi) is zoo assert cos(3.0*o*pi) == -1 assert sin(3.0*o*pi) == 0 assert tan(3.0*o*pi) == 0 assert cot(3.0*o*pi) is zoo assert cos(4.5*o*pi) == cos(0.5*pi*o) assert sin(4.5*o*pi) == sin(0.5*pi*o) assert tan(4.5*o*pi) == tan(0.5*pi*o) assert cot(4.5*o*pi) == cot(0.5*pi*o) # x could be imaginary assert cos(4*x*pi) == cos(4*pi*x) assert sin(4*x*pi) == sin(4*pi*x) assert tan(4*x*pi) == tan(4*pi*x) assert cot(4*x*pi) == cot(4*pi*x) assert cos(3*x*pi) == cos(3*pi*x) assert sin(3*x*pi) == sin(3*pi*x) assert tan(3*x*pi) == tan(3*pi*x) assert cot(3*x*pi) == cot(3*pi*x) assert cos(4.0*x*pi) == cos(4.0*pi*x) assert sin(4.0*x*pi) == sin(4.0*pi*x) assert tan(4.0*x*pi) == tan(4.0*pi*x) assert cot(4.0*x*pi) == cot(4.0*pi*x) assert cos(3.0*x*pi) == cos(3.0*pi*x) assert sin(3.0*x*pi) == sin(3.0*pi*x) assert tan(3.0*x*pi) == tan(3.0*pi*x) assert cot(3.0*x*pi) == cot(3.0*pi*x) assert cos(4.5*x*pi) == cos(4.5*pi*x) assert sin(4.5*x*pi) == sin(4.5*pi*x) assert tan(4.5*x*pi) == tan(4.5*pi*x) assert cot(4.5*x*pi) == cot(4.5*pi*x) def test_inverses(): raises(AttributeError, lambda: sin(x).inverse()) raises(AttributeError, lambda: cos(x).inverse()) assert tan(x).inverse() == atan assert cot(x).inverse() == acot raises(AttributeError, lambda: csc(x).inverse()) raises(AttributeError, lambda: sec(x).inverse()) assert asin(x).inverse() == sin assert acos(x).inverse() == cos assert atan(x).inverse() == tan assert acot(x).inverse() == cot def test_real_imag(): a, b = symbols('a b', real=True) z = a + b*I for deep in [True, False]: assert sin( z).as_real_imag(deep=deep) == (sin(a)*cosh(b), cos(a)*sinh(b)) assert cos( z).as_real_imag(deep=deep) == (cos(a)*cosh(b), -sin(a)*sinh(b)) assert tan(z).as_real_imag(deep=deep) == (sin(2*a)/(cos(2*a) + cosh(2*b)), sinh(2*b)/(cos(2*a) + cosh(2*b))) assert cot(z).as_real_imag(deep=deep) == (-sin(2*a)/(cos(2*a) - cosh(2*b)), sinh(2*b)/(cos(2*a) - cosh(2*b))) assert sin(a).as_real_imag(deep=deep) == (sin(a), 0) assert cos(a).as_real_imag(deep=deep) == (cos(a), 0) assert tan(a).as_real_imag(deep=deep) == (tan(a), 0) assert cot(a).as_real_imag(deep=deep) == (cot(a), 0) @XFAIL def test_sin_cos_with_infinity(): # Test for issue 5196 # https://github.com/sympy/sympy/issues/5196 assert sin(oo) is S.NaN assert cos(oo) is S.NaN @slow def test_sincos_rewrite_sqrt(): # equivalent to testing rewrite(pow) for p in [1, 3, 5, 17]: for t in [1, 8]: n = t*p # The vertices `exp(i*pi/n)` of a regular `n`-gon can # be expressed by means of nested square roots if and # only if `n` is a product of Fermat primes, `p`, and # powers of 2, `t'. The code aims to check all vertices # not belonging to an `m`-gon for `m < n`(`gcd(i, n) == 1`). # For large `n` this makes the test too slow, therefore # the vertices are limited to those of index `i < 10`. for i in range(1, min((n + 1)//2 + 1, 10)): if 1 == gcd(i, n): x = i*pi/n s1 = sin(x).rewrite(sqrt) c1 = cos(x).rewrite(sqrt) assert not s1.has(cos, sin), "fails for %d*pi/%d" % (i, n) assert not c1.has(cos, sin), "fails for %d*pi/%d" % (i, n) assert 1e-3 > abs(sin(x.evalf(5)) - s1.evalf(2)), "fails for %d*pi/%d" % (i, n) assert 1e-3 > abs(cos(x.evalf(5)) - c1.evalf(2)), "fails for %d*pi/%d" % (i, n) assert cos(pi/14).rewrite(sqrt) == sqrt(cos(pi/7)/2 + S.Half) assert cos(pi/257).rewrite(sqrt).evalf(64) == cos(pi/257).evalf(64) assert cos(pi*Rational(-15, 2)/11, evaluate=False).rewrite( sqrt) == -sqrt(-cos(pi*Rational(4, 11))/2 + S.Half) assert cos(Mul(2, pi, S.Half, evaluate=False), evaluate=False).rewrite( sqrt) == -1 e = cos(pi/3/17) # don't use pi/15 since that is caught at instantiation a = ( -3*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17) + 17)/64 - 3*sqrt(34)*sqrt(sqrt(17) + 17)/128 - sqrt(sqrt(17) + 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - sqrt(-sqrt(17) + 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 - Rational(1, 32) + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 + 3*sqrt(2)*sqrt(sqrt(17) + 17)/128 + sqrt(34)*sqrt(-sqrt(17) + 17)/128 + 13*sqrt(2)*sqrt(-sqrt(17) + 17)/128 + sqrt(17)*sqrt(-sqrt(17) + 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 + 5*sqrt(17)/32 + sqrt(3)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/8 - 5*sqrt(2)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - 3*sqrt(2)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/32 + sqrt(34)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 + sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/2 + S.Half + sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + sqrt(34)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/32)/2) assert e.rewrite(sqrt) == a assert e.n() == a.n() # coverage of fermatCoords: multiplicity > 1; the following could be # different but that portion of the code should be tested in some way assert cos(pi/9/17).rewrite(sqrt) == \ sin(pi/9)*sin(pi*Rational(2, 17)) + cos(pi/9)*cos(pi*Rational(2, 17)) @slow def test_tancot_rewrite_sqrt(): # equivalent to testing rewrite(pow) for p in [1, 3, 5, 17]: for t in [1, 8]: n = t*p for i in range(1, min((n + 1)//2 + 1, 10)): if 1 == gcd(i, n): x = i*pi/n if 2*i != n and 3*i != 2*n: t1 = tan(x).rewrite(sqrt) assert not t1.has(cot, tan), "fails for %d*pi/%d" % (i, n) assert 1e-3 > abs( tan(x.evalf(7)) - t1.evalf(4) ), "fails for %d*pi/%d" % (i, n) if i != 0 and i != n: c1 = cot(x).rewrite(sqrt) assert not c1.has(cot, tan), "fails for %d*pi/%d" % (i, n) assert 1e-3 > abs( cot(x.evalf(7)) - c1.evalf(4) ), "fails for %d*pi/%d" % (i, n) def test_sec(): x = symbols('x', real=True) z = symbols('z') assert sec.nargs == FiniteSet(1) assert sec(zoo) is nan assert sec(0) == 1 assert sec(pi) == -1 assert sec(pi/2) is zoo assert sec(-pi/2) is zoo assert sec(pi/6) == 2*sqrt(3)/3 assert sec(pi/3) == 2 assert sec(pi*Rational(5, 2)) is zoo assert sec(pi*Rational(9, 7)) == -sec(pi*Rational(2, 7)) assert sec(pi*Rational(3, 4)) == -sqrt(2) # issue 8421 assert sec(I) == 1/cosh(1) assert sec(x*I) == 1/cosh(x) assert sec(-x) == sec(x) assert sec(asec(x)) == x assert sec(z).conjugate() == sec(conjugate(z)) assert (sec(z).as_real_imag() == (cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2), sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 + cos(re(z))**2*cosh(im(z))**2))) assert sec(x).expand(trig=True) == 1/cos(x) assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1) assert sec(x).is_extended_real == True assert sec(z).is_real == None assert sec(a).is_algebraic is None assert sec(na).is_algebraic is False assert sec(x).as_leading_term() == sec(x) assert sec(0, evaluate=False).is_finite == True assert sec(x).is_finite == None assert sec(pi/2, evaluate=False).is_finite == False assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6) # https://github.com/sympy/sympy/issues/7166 assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6) # https://github.com/sympy/sympy/issues/7167 assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) == 1/sqrt(x - pi*Rational(3, 2)) + (x - pi*Rational(3, 2))**Rational(3, 2)/12 + (x - pi*Rational(3, 2))**Rational(7, 2)/160 + O((x - pi*Rational(3, 2))**4, (x, pi*Rational(3, 2)))) assert sec(x).diff(x) == tan(x)*sec(x) # Taylor Term checks assert sec(z).taylor_term(4, z) == 5*z**4/24 assert sec(z).taylor_term(6, z) == 61*z**6/720 assert sec(z).taylor_term(5, z) == 0 def test_sec_rewrite(): assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2) assert sec(x).rewrite(cos) == 1/cos(x) assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1) assert sec(x).rewrite(pow) == sec(x) assert sec(x).rewrite(sqrt) == sec(x) assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1) assert sec(x).rewrite(sin) == 1 / sin(x + pi / 2, evaluate=False) assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1) assert sec(x).rewrite(csc) == csc(-x + pi/2, evaluate=False) def test_sec_fdiff(): assert sec(x).fdiff() == tan(x)*sec(x) raises(ArgumentIndexError, lambda: sec(x).fdiff(2)) def test_csc(): x = symbols('x', real=True) z = symbols('z') # https://github.com/sympy/sympy/issues/6707 cosecant = csc('x') alternate = 1/sin('x') assert cosecant.equals(alternate) == True assert alternate.equals(cosecant) == True assert csc.nargs == FiniteSet(1) assert csc(0) is zoo assert csc(pi) is zoo assert csc(zoo) is nan assert csc(pi/2) == 1 assert csc(-pi/2) == -1 assert csc(pi/6) == 2 assert csc(pi/3) == 2*sqrt(3)/3 assert csc(pi*Rational(5, 2)) == 1 assert csc(pi*Rational(9, 7)) == -csc(pi*Rational(2, 7)) assert csc(pi*Rational(3, 4)) == sqrt(2) # issue 8421 assert csc(I) == -I/sinh(1) assert csc(x*I) == -I/sinh(x) assert csc(-x) == -csc(x) assert csc(acsc(x)) == x assert csc(z).conjugate() == csc(conjugate(z)) assert (csc(z).as_real_imag() == (sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 + cos(re(z))**2*sinh(im(z))**2), -cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 + cos(re(z))**2*sinh(im(z))**2))) assert csc(x).expand(trig=True) == 1/sin(x) assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x)) assert csc(x).is_extended_real == True assert csc(z).is_real == None assert csc(a).is_algebraic is None assert csc(na).is_algebraic is False assert csc(x).as_leading_term() == csc(x) assert csc(0, evaluate=False).is_finite == False assert csc(x).is_finite == None assert csc(pi/2, evaluate=False).is_finite == True assert series(csc(x), x, x0=pi/2, n=6) == \ 1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2)) assert series(csc(x), x, x0=0, n=6) == \ 1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6) assert csc(x).diff(x) == -cot(x)*csc(x) assert csc(x).taylor_term(2, x) == 0 assert csc(x).taylor_term(3, x) == 7*x**3/360 assert csc(x).taylor_term(5, x) == 31*x**5/15120 raises(ArgumentIndexError, lambda: csc(x).fdiff(2)) def test_asec(): z = Symbol('z', zero=True) assert asec(z) is zoo assert asec(nan) is nan assert asec(1) == 0 assert asec(-1) == pi assert asec(oo) == pi/2 assert asec(-oo) == pi/2 assert asec(zoo) == pi/2 assert asec(sec(pi*Rational(13, 4))) == pi*Rational(3, 4) assert asec(1 + sqrt(5)) == pi*Rational(2, 5) assert asec(2/sqrt(3)) == pi/6 assert asec(sqrt(4 - 2*sqrt(2))) == pi/8 assert asec(-sqrt(4 + 2*sqrt(2))) == pi*Rational(5, 8) assert asec(sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(3, 10) assert asec(-sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(7, 10) assert asec(sqrt(2) - sqrt(6)) == pi*Rational(11, 12) assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2)) assert asec(x).as_leading_term(x) == I*log(x) assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2 assert asec(x).rewrite(asin) == -asin(1/x) + pi/2 assert asec(x).rewrite(acos) == acos(1/x) assert asec(x).rewrite(atan) == (2*atan(x + sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x assert asec(x).rewrite(acot) == (2*acot(x - sqrt(x**2 - 1)) - pi/2)*sqrt(x**2)/x assert asec(x).rewrite(acsc) == -acsc(x) + pi/2 raises(ArgumentIndexError, lambda: asec(x).fdiff(2)) def test_asec_is_real(): assert asec(S.Half).is_real is False n = Symbol('n', positive=True, integer=True) assert asec(n).is_extended_real is True assert asec(x).is_real is None assert asec(r).is_real is None t = Symbol('t', real=False, finite=True) assert asec(t).is_real is False def test_acsc(): assert acsc(nan) is nan assert acsc(1) == pi/2 assert acsc(-1) == -pi/2 assert acsc(oo) == 0 assert acsc(-oo) == 0 assert acsc(zoo) == 0 assert acsc(0) is zoo assert acsc(csc(3)) == -3 + pi assert acsc(csc(4)) == -4 + pi assert acsc(csc(6)) == 6 - 2*pi assert unchanged(acsc, csc(x)) assert unchanged(acsc, sec(x)) assert acsc(2/sqrt(3)) == pi/3 assert acsc(csc(pi*Rational(13, 4))) == -pi/4 assert acsc(sqrt(2 + 2*sqrt(5)/5)) == pi/5 assert acsc(-sqrt(2 + 2*sqrt(5)/5)) == -pi/5 assert acsc(-2) == -pi/6 assert acsc(-sqrt(4 + 2*sqrt(2))) == -pi/8 assert acsc(sqrt(4 - 2*sqrt(2))) == pi*Rational(3, 8) assert acsc(1 + sqrt(5)) == pi/10 assert acsc(sqrt(2) - sqrt(6)) == pi*Rational(-5, 12) assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2)) assert acsc(x).as_leading_term(x) == I*log(x) assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x) assert acsc(x).rewrite(asin) == asin(1/x) assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2 assert acsc(x).rewrite(atan) == (-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x assert acsc(x).rewrite(asec) == -asec(x) + pi/2 raises(ArgumentIndexError, lambda: acsc(x).fdiff(2)) def test_csc_rewrite(): assert csc(x).rewrite(pow) == csc(x) assert csc(x).rewrite(sqrt) == csc(x) assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x)) assert csc(x).rewrite(sin) == 1/sin(x) assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2)) assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2)) assert csc(x).rewrite(cos) == 1/cos(x - pi/2, evaluate=False) assert csc(x).rewrite(sec) == sec(-x + pi/2, evaluate=False) # issue 17349 assert csc(1 - exp(-besselj(I, I))).rewrite(cos) == \ -1/cos(-pi/2 - 1 + cos(I*besselj(I, I)) + I*cos(-pi/2 + I*besselj(I, I), evaluate=False), evaluate=False) def test_inverses_nseries(): assert asin(x + 2)._eval_nseries(x, 4, None, I) == -asin(2) + pi + sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + \ sqrt(3)*I*x**3/18 + O(x**4) assert asin(x + 2)._eval_nseries(x, 4, None, -I) == asin(2) - sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4) assert asin(x - 2)._eval_nseries(x, 4, None, I) == -asin(2) - sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4) assert asin(x - 2)._eval_nseries(x, 4, None, -I) == asin(2) - pi + sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + \ sqrt(3)*I*x**3/18 + O(x**4) assert asin(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -asin(2) + pi - sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3) assert asin(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == asin(2) + sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3) assert asin(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == -asin(2) + sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3) assert asin(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == asin(2) - pi - sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3) assert asin(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -asin(2) + pi - sqrt(3)*x**2/3 + O(x**3) assert asin(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == -asin(2) + pi - sqrt(3)*x**2/3 + O(x**3) assert asin(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == -asin(2) + sqrt(3)*x**2/3 + O(x**3) assert asin(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == -asin(2) + sqrt(3)*x**2/3 + O(x**3) assert asin(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) - \ sqrt(2)*(-x)**(S(3)/2)/12 - 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3) assert asin(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) + \ sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3) assert asin(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) + \ sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3) assert asin(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \ sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3) assert acos(x + 2)._eval_nseries(x, 4, None, I) == -acos(2) - sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4) assert acos(x + 2)._eval_nseries(x, 4, None, -I) == acos(2) + sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4) assert acos(x - 2)._eval_nseries(x, 4, None, I) == acos(-2) + sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4) assert acos(x - 2)._eval_nseries(x, 4, None, -I) == -acos(-2) + 2*pi - sqrt(3)*I*x/3 - \ sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4) # assert acos(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -acos(2) + sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3) # assert acos(I*x + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == acos(2) - sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3) # assert acos(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == acos(-2) - sqrt(3)*x/3 - sqrt(3)*I*x**2/9 + O(x**3) # assert acos(I*x + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == -acos(-2) + 2*pi + sqrt(3)*x/3 + sqrt(3)*I*x**2/9 + O(x**3) # assert acos(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, 1) == -acos(2) + sqrt(3)*x**2/3 + O(x**3) # assert acos(I*x**2 + I*x**3 + 2)._eval_nseries(x, 3, None, -1) == -acos(2) + sqrt(3)*x**2/3 + O(x**3) # assert acos(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, 1) == acos(-2) - sqrt(3)*x**2/3 + O(x**3) # assert acos(I*x**2 + I*x**3 - 2)._eval_nseries(x, 3, None, -1) == acos(-2) - sqrt(3)*x**2/3 + O(x**3) # assert acos(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) + sqrt(2)*(-x)**(S(3)/2)/12 + 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3) # assert acos(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/12 - 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3) # assert acos(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) - sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3) # assert acos(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3) assert atan(x + 2*I)._eval_nseries(x, 4, None, 1) == I*atanh(2) - x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4) assert atan(x + 2*I)._eval_nseries(x, 4, None, -1) == I*atanh(2) - pi - x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4) assert atan(x - 2*I)._eval_nseries(x, 4, None, 1) == -I*atanh(2) + pi - x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4) assert atan(x - 2*I)._eval_nseries(x, 4, None, -1) == -I*atanh(2) - x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4) # assert atan(x**2 + 2*I)._eval_nseries(x, 3, None, 1) == I*atanh(2) - x**2/3 + O(x**3) # assert atan(x**2 + 2*I)._eval_nseries(x, 3, None, -1) == I*atanh(2) - x**2/3 + O(x**3) # assert atan(x**2 - 2*I)._eval_nseries(x, 3, None, 1) == -I*atanh(2) + pi - x**2/3 + O(x**3) # assert atan(x**2 - 2*I)._eval_nseries(x, 3, None, -1) == -I*atanh(2) + pi - x**2/3 + O(x**3) assert atan(1/x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2) assert atan(1/x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2) assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, 1) == -I*acoth(S(1)/2) + pi - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4) assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, -1) == -I*acoth(S(1)/2) - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4) assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, 1) == I*acoth(S(1)/2) - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4) assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, -1) == I*acoth(S(1)/2) - pi - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4) # assert acot(x**2 + S(1)/2*I)._eval_nseries(x, 3, None, 1) == -I*acoth(S(1)/2) + pi - 4*x**2/3 + O(x**3) # assert acot(x**2 + S(1)/2*I)._eval_nseries(x, 3, None, -1) == -I*acoth(S(1)/2) + pi - 4*x**2/3 + O(x**3) # assert acot(x**2 - S(1)/2*I)._eval_nseries(x, 3, None, 1) == I*acoth(S(1)/2) - 4*x**2/3 + O(x**3) # assert acot(x**2 - S(1)/2*I)._eval_nseries(x, 3, None, -1) == I*acoth(S(1)/2) - 4*x**2/3 + O(x**3) # assert acot(x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2) # assert acot(x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2) assert asec(x + S(1)/2)._eval_nseries(x, 4, None, I) == asec(S(1)/2) - 4*sqrt(3)*I*x/3 + \ 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4) assert asec(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -asec(S(1)/2) + 4*sqrt(3)*I*x/3 - \ 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4) assert asec(x - S(1)/2)._eval_nseries(x, 4, None, I) == -asec(-S(1)/2) + 2*pi + 4*sqrt(3)*I*x/3 + \ 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4) assert asec(x - S(1)/2)._eval_nseries(x, 4, None, -I) == asec(-S(1)/2) - 4*sqrt(3)*I*x/3 - \ 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4) # assert asec(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == asec(S(1)/2) + 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3) # assert asec(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == -asec(S(1)/2) - 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3) # assert asec(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3) # assert asec(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == asec(-S(1)/2) + 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3) # assert asec(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == asec(S(1)/2) + 4*sqrt(3)*x**2/3 + O(x**3) # assert asec(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == asec(S(1)/2) + 4*sqrt(3)*x**2/3 + O(x**3) # assert asec(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x**2/3 + O(x**3) # assert asec(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == -asec(-S(1)/2) + 2*pi - 4*sqrt(3)*x**2/3 + O(x**3) # assert asec(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - 5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3) # assert asec(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + 5*sqrt(2)*(-x)**(S(3)/2)/12 - 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3) # assert asec(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3) # assert asec(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) + sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3) assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) + 4*sqrt(3)*I*x/3 - \ 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4) assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + pi - 4*sqrt(3)*I*x/3 + \ 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4) assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) - pi - 4*sqrt(3)*I*x/3 - \ 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4) assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + 4*sqrt(3)*I*x/3 + \ 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4) # assert acsc(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3) # assert acsc(I*x + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == -acsc(S(1)/2) + pi + 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3) # assert acsc(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x/3 + 8*sqrt(3)*I*x**2/9 + O(x**3) # assert acsc(I*x + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == -acsc(S(1)/2) - 4*sqrt(3)*x/3 - 8*sqrt(3)*I*x**2/9 + O(x**3) # assert acsc(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - 4*sqrt(3)*x**2/3 + O(x**3) # assert acsc(I*x**2 + I*x**3 + S(1)/2)._eval_nseries(x, 3, None, -1) == acsc(S(1)/2) - 4*sqrt(3)*x**2/3 + O(x**3) # assert acsc(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, 1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x**2/3 + O(x**3) # assert acsc(I*x**2 + I*x**3 - S(1)/2)._eval_nseries(x, 3, None, -1) == acsc(S(1)/2) - pi + 4*sqrt(3)*x**2/3 + O(x**3) # assert acsc(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + 5*sqrt(2)*x**(S(3)/2)/12 - 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3) # assert acsc(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - 5*sqrt(2)*(-x)**(S(3)/2)/12 + 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3) # assert acsc(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3) # assert acsc(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) - sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3) def test_issue_8653(): n = Symbol('n', integer=True) assert sin(n).is_irrational is None assert cos(n).is_irrational is None assert tan(n).is_irrational is None def test_issue_9157(): n = Symbol('n', integer=True, positive=True) assert atan(n - 1).is_nonnegative is True def test_trig_period(): x, y = symbols('x, y') assert sin(x).period() == 2*pi assert cos(x).period() == 2*pi assert tan(x).period() == pi assert cot(x).period() == pi assert sec(x).period() == 2*pi assert csc(x).period() == 2*pi assert sin(2*x).period() == pi assert cot(4*x - 6).period() == pi/4 assert cos((-3)*x).period() == pi*Rational(2, 3) assert cos(x*y).period(x) == 2*pi/abs(y) assert sin(3*x*y + 2*pi).period(y) == 2*pi/abs(3*x) assert tan(3*x).period(y) is S.Zero raises(NotImplementedError, lambda: sin(x**2).period(x)) def test_issue_7171(): assert sin(x).rewrite(sqrt) == sin(x) assert sin(x).rewrite(pow) == sin(x) def test_issue_11864(): w, k = symbols('w, k', real=True) F = Piecewise((1, Eq(2*pi*k, 0)), (sin(pi*k)/(pi*k), True)) soln = Piecewise((1, Eq(2*pi*k, 0)), (sinc(pi*k), True)) assert F.rewrite(sinc) == soln def test_real_assumptions(): z = Symbol('z', real=False, finite=True) assert sin(z).is_real is None assert cos(z).is_real is None assert tan(z).is_real is False assert sec(z).is_real is None assert csc(z).is_real is None assert cot(z).is_real is False assert asin(p).is_real is None assert asin(n).is_real is None assert asec(p).is_real is None assert asec(n).is_real is None assert acos(p).is_real is None assert acos(n).is_real is None assert acsc(p).is_real is None assert acsc(n).is_real is None assert atan(p).is_positive is True assert atan(n).is_negative is True assert acot(p).is_positive is True assert acot(n).is_negative is True def test_issue_14320(): assert asin(sin(2)) == -2 + pi and (-pi/2 <= -2 + pi <= pi/2) and sin(2) == sin(-2 + pi) assert asin(cos(2)) == -2 + pi/2 and (-pi/2 <= -2 + pi/2 <= pi/2) and cos(2) == sin(-2 + pi/2) assert acos(sin(2)) == -pi/2 + 2 and (0 <= -pi/2 + 2 <= pi) and sin(2) == cos(-pi/2 + 2) assert acos(cos(20)) == -6*pi + 20 and (0 <= -6*pi + 20 <= pi) and cos(20) == cos(-6*pi + 20) assert acos(cos(30)) == -30 + 10*pi and (0 <= -30 + 10*pi <= pi) and cos(30) == cos(-30 + 10*pi) assert atan(tan(17)) == -5*pi + 17 and (-pi/2 < -5*pi + 17 < pi/2) and tan(17) == tan(-5*pi + 17) assert atan(tan(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 < pi/2) and tan(15) == tan(-5*pi + 15) assert atan(cot(12)) == -12 + pi*Rational(7, 2) and (-pi/2 < -12 + pi*Rational(7, 2) < pi/2) and cot(12) == tan(-12 + pi*Rational(7, 2)) assert acot(cot(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 <= pi/2) and cot(15) == cot(-5*pi + 15) assert acot(tan(19)) == -19 + pi*Rational(13, 2) and (-pi/2 < -19 + pi*Rational(13, 2) <= pi/2) and tan(19) == cot(-19 + pi*Rational(13, 2)) assert asec(sec(11)) == -11 + 4*pi and (0 <= -11 + 4*pi <= pi) and cos(11) == cos(-11 + 4*pi) assert asec(csc(13)) == -13 + pi*Rational(9, 2) and (0 <= -13 + pi*Rational(9, 2) <= pi) and sin(13) == cos(-13 + pi*Rational(9, 2)) assert acsc(csc(14)) == -4*pi + 14 and (-pi/2 <= -4*pi + 14 <= pi/2) and sin(14) == sin(-4*pi + 14) assert acsc(sec(10)) == pi*Rational(-7, 2) + 10 and (-pi/2 <= pi*Rational(-7, 2) + 10 <= pi/2) and cos(10) == sin(pi*Rational(-7, 2) + 10) def test_issue_14543(): assert sec(2*pi + 11) == sec(11) assert sec(2*pi - 11) == sec(11) assert sec(pi + 11) == -sec(11) assert sec(pi - 11) == -sec(11) assert csc(2*pi + 17) == csc(17) assert csc(2*pi - 17) == -csc(17) assert csc(pi + 17) == -csc(17) assert csc(pi - 17) == csc(17) x = Symbol('x') assert csc(pi/2 + x) == sec(x) assert csc(pi/2 - x) == sec(x) assert csc(pi*Rational(3, 2) + x) == -sec(x) assert csc(pi*Rational(3, 2) - x) == -sec(x) assert sec(pi/2 - x) == csc(x) assert sec(pi/2 + x) == -csc(x) assert sec(pi*Rational(3, 2) + x) == csc(x) assert sec(pi*Rational(3, 2) - x) == -csc(x) def test_as_real_imag(): # This is for https://github.com/sympy/sympy/issues/17142 # If it start failing again in irrelevant builds or in the master # please open up the issue again. expr = atan(I/(I + I*tan(1))) assert expr.as_real_imag() == (expr, 0) def test_issue_18746(): e3 = cos(S.Pi*(x/4 + 1/4)) assert e3.period() == 8