def test_math_functions():
dort('sin(37)', sympy.sin(37))
dort('cos(38)', sympy.cos(38))
dort('tan(38)', sympy.tan(38))
dort('sec(39)', sympy.sec(39))
dort('csc(40)', sympy.csc(40))
dort('cot(41)', sympy.cot(41))
dort('asin(42)', sympy.asin(42))
dort('acos(43)', sympy.acos(43))
dort('atan(44)', sympy.atan(44))
dort('asec(45)', sympy.asec(45))
dort('acsc(46)', sympy.acsc(46))
dort('acot(47)', sympy.acot(47))
dort('sind(37)', sympy.sin(37 * sympy.pi / sympy.Number(180)))
dort('cosd(38)', sympy.cos(38 * sympy.pi / sympy.Number(180)))
dort('tand(38)', sympy.tan(38 * sympy.pi / sympy.Number(180)))
dort('secd(39)', sympy.sec(39 * sympy.pi / sympy.Number(180)))
dort('cscd(40)', sympy.csc(40 * sympy.pi / sympy.Number(180)))
dort('cotd(41)', sympy.cot(41 * sympy.pi / sympy.Number(180)))
dort('asind(42)', sympy.asin(42) * sympy.Number(180) / sympy.pi)
dort('acosd(43)', sympy.acos(43) * sympy.Number(180) / sympy.pi)
dort('atand(44)', sympy.atan(44) * sympy.Number(180) / sympy.pi)
dort('asecd(45)', sympy.asec(45) * sympy.Number(180) / sympy.pi)
dort('acscd(46)', sympy.acsc(46) * sympy.Number(180) / sympy.pi)
dort('acotd(47)', sympy.acot(47) * sympy.Number(180) / sympy.pi)
dort('sinh(4)', sympy.sinh(4))
dort('cosh(5)', sympy.cosh(5))
dort('tanh(6)', sympy.tanh(6))
dort('asinh(4)', sympy.asinh(4))
dort('acosh(5)', sympy.acosh(5))
dort('atanh(6)', sympy.atanh(6))
dort('int(E)', int(sympy.E))
dort('int(-E)', int(-sympy.E))
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_acot_rewrite():
assert acot(x).rewrite(log) == I*log((x - I)/(x + I))/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))
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_acot_rewrite():
assert acot(x).rewrite(log) == I*log((x - I)/(x + I))/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))
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*log((1 - I*x)/(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
def test_conv7():
x = Symbol("x")
y = Symbol("y")
assert sin(x/3) == sin(sympy.Symbol("x") / 3)
assert cos(x/3) == cos(sympy.Symbol("x") / 3)
assert tan(x/3) == tan(sympy.Symbol("x") / 3)
assert cot(x/3) == cot(sympy.Symbol("x") / 3)
assert csc(x/3) == csc(sympy.Symbol("x") / 3)
assert sec(x/3) == sec(sympy.Symbol("x") / 3)
assert asin(x/3) == asin(sympy.Symbol("x") / 3)
assert acos(x/3) == acos(sympy.Symbol("x") / 3)
assert atan(x/3) == atan(sympy.Symbol("x") / 3)
assert acot(x/3) == acot(sympy.Symbol("x") / 3)
assert acsc(x/3) == acsc(sympy.Symbol("x") / 3)
assert asec(x/3) == asec(sympy.Symbol("x") / 3)
assert sin(x/3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3)
assert sin(x/3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3)
assert cos(x/3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3)
assert tan(x/3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3)
assert cot(x/3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3)
assert csc(x/3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3)
assert sec(x/3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3)
assert asin(x/3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3)
assert acos(x/3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3)
assert atan(x/3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3)
assert acot(x/3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3)
assert acsc(x/3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3)
assert asec(x/3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)
def test_trig_functions(self, printer, x):
# Trig functions
assert printer.doprint(sp.acos(x)) == 'acos(x)'
assert printer.doprint(sp.acosh(x)) == 'acosh(x)'
assert printer.doprint(sp.asin(x)) == 'asin(x)'
assert printer.doprint(sp.asinh(x)) == 'asinh(x)'
assert printer.doprint(sp.atan(x)) == 'atan(x)'
assert printer.doprint(sp.atanh(x)) == 'atanh(x)'
assert printer.doprint(sp.ceiling(x)) == 'ceil(x)'
assert printer.doprint(sp.cos(x)) == 'cos(x)'
assert printer.doprint(sp.cosh(x)) == 'cosh(x)'
assert printer.doprint(sp.exp(x)) == 'exp(x)'
assert printer.doprint(sp.factorial(x)) == 'factorial(x)'
assert printer.doprint(sp.floor(x)) == 'floor(x)'
assert printer.doprint(sp.log(x)) == 'log(x)'
assert printer.doprint(sp.sin(x)) == 'sin(x)'
assert printer.doprint(sp.sinh(x)) == 'sinh(x)'
assert printer.doprint(sp.tan(x)) == 'tan(x)'
assert printer.doprint(sp.tanh(x)) == 'tanh(x)'
# extra trig functions
assert printer.doprint(sp.sec(x)) == '1 / cos(x)'
assert printer.doprint(sp.csc(x)) == '1 / sin(x)'
assert printer.doprint(sp.cot(x)) == '1 / tan(x)'
assert printer.doprint(sp.asec(x)) == 'acos(1 / x)'
assert printer.doprint(sp.acsc(x)) == 'asin(1 / x)'
assert printer.doprint(sp.acot(x)) == 'atan(1 / x)'
assert printer.doprint(sp.sech(x)) == '1 / cosh(x)'
assert printer.doprint(sp.csch(x)) == '1 / sinh(x)'
assert printer.doprint(sp.coth(x)) == '1 / tanh(x)'
assert printer.doprint(sp.asech(x)) == 'acosh(1 / x)'
assert printer.doprint(sp.acsch(x)) == 'asinh(1 / x)'
assert printer.doprint(sp.acoth(x)) == 'atanh(1 / x)'
def test_conv7():
x = Symbol("x")
y = Symbol("y")
assert sin(x / 3) == sin(sympy.Symbol("x") / 3)
assert cos(x / 3) == cos(sympy.Symbol("x") / 3)
assert tan(x / 3) == tan(sympy.Symbol("x") / 3)
assert cot(x / 3) == cot(sympy.Symbol("x") / 3)
assert csc(x / 3) == csc(sympy.Symbol("x") / 3)
assert sec(x / 3) == sec(sympy.Symbol("x") / 3)
assert asin(x / 3) == asin(sympy.Symbol("x") / 3)
assert acos(x / 3) == acos(sympy.Symbol("x") / 3)
assert atan(x / 3) == atan(sympy.Symbol("x") / 3)
assert acot(x / 3) == acot(sympy.Symbol("x") / 3)
assert acsc(x / 3) == acsc(sympy.Symbol("x") / 3)
assert asec(x / 3) == asec(sympy.Symbol("x") / 3)
assert sin(x / 3)._sympy_() == sympy.sin(sympy.Symbol("x") / 3)
assert sin(x / 3)._sympy_() != sympy.cos(sympy.Symbol("x") / 3)
assert cos(x / 3)._sympy_() == sympy.cos(sympy.Symbol("x") / 3)
assert tan(x / 3)._sympy_() == sympy.tan(sympy.Symbol("x") / 3)
assert cot(x / 3)._sympy_() == sympy.cot(sympy.Symbol("x") / 3)
assert csc(x / 3)._sympy_() == sympy.csc(sympy.Symbol("x") / 3)
assert sec(x / 3)._sympy_() == sympy.sec(sympy.Symbol("x") / 3)
assert asin(x / 3)._sympy_() == sympy.asin(sympy.Symbol("x") / 3)
assert acos(x / 3)._sympy_() == sympy.acos(sympy.Symbol("x") / 3)
assert atan(x / 3)._sympy_() == sympy.atan(sympy.Symbol("x") / 3)
assert acot(x / 3)._sympy_() == sympy.acot(sympy.Symbol("x") / 3)
assert acsc(x / 3)._sympy_() == sympy.acsc(sympy.Symbol("x") / 3)
assert asec(x / 3)._sympy_() == sympy.asec(sympy.Symbol("x") / 3)
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*log((1 - I*x)/(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
def acsc(self, value):
"""
Verilen değeri kullanarak, sumpy.acsc döndürür
:param value: değer
:return: sympy.acsc
"""
return sp.acsc(value_checker(value))
def __new__(cls, arg):
obj = acsc(arg)
if arg.is_real:
assumptions = {'real': True}
ass_copy = assumptions.copy()
obj._assumptions = StdFactKB(assumptions)
obj._assumptions._generator = ass_copy
return obj
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_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_branch_cuts():
assert limit(asin(I * x + 2), x, 0) == pi - asin(2)
assert limit(asin(I * x + 2), x, 0, '-') == asin(2)
assert limit(asin(I * x - 2), x, 0) == -asin(2)
assert limit(asin(I * x - 2), x, 0, '-') == -pi + asin(2)
assert limit(acos(I * x + 2), x, 0) == -acos(2)
assert limit(acos(I * x + 2), x, 0, '-') == acos(2)
assert limit(acos(I * x - 2), x, 0) == acos(-2)
assert limit(acos(I * x - 2), x, 0, '-') == 2 * pi - acos(-2)
assert limit(atan(x + 2 * I), x, 0) == I * atanh(2)
assert limit(atan(x + 2 * I), x, 0, '-') == -pi + I * atanh(2)
assert limit(atan(x - 2 * I), x, 0) == pi - I * atanh(2)
assert limit(atan(x - 2 * I), x, 0, '-') == -I * atanh(2)
assert limit(atan(1 / x), x, 0) == pi / 2
assert limit(atan(1 / x), x, 0, '-') == -pi / 2
assert limit(atan(x), x, oo) == pi / 2
assert limit(atan(x), x, -oo) == -pi / 2
assert limit(acot(x + S(1) / 2 * I), x, 0) == pi - I * acoth(S(1) / 2)
assert limit(acot(x + S(1) / 2 * I), x, 0, '-') == -I * acoth(S(1) / 2)
assert limit(acot(x - S(1) / 2 * I), x, 0) == I * acoth(S(1) / 2)
assert limit(acot(x - S(1) / 2 * I), x, 0,
'-') == -pi + I * acoth(S(1) / 2)
assert limit(acot(x), x, 0) == pi / 2
assert limit(acot(x), x, 0, '-') == -pi / 2
assert limit(asec(I * x + S(1) / 2), x, 0) == asec(S(1) / 2)
assert limit(asec(I * x + S(1) / 2), x, 0, '-') == -asec(S(1) / 2)
assert limit(asec(I * x - S(1) / 2), x, 0) == 2 * pi - asec(-S(1) / 2)
assert limit(asec(I * x - S(1) / 2), x, 0, '-') == asec(-S(1) / 2)
assert limit(acsc(I * x + S(1) / 2), x, 0) == acsc(S(1) / 2)
assert limit(acsc(I * x + S(1) / 2), x, 0, '-') == pi - acsc(S(1) / 2)
assert limit(acsc(I * x - S(1) / 2), x, 0) == -pi + acsc(S(1) / 2)
assert limit(acsc(I * x - S(1) / 2), x, 0, '-') == -acsc(S(1) / 2)
assert limit(log(I * x - 1), x, 0) == I * pi
assert limit(log(I * x - 1), x, 0, '-') == -I * pi
assert limit(log(-I * x - 1), x, 0) == -I * pi
assert limit(log(-I * x - 1), x, 0, '-') == I * pi
assert limit(sqrt(I * x - 1), x, 0) == I
assert limit(sqrt(I * x - 1), x, 0, '-') == -I
assert limit(sqrt(-I * x - 1), x, 0) == -I
assert limit(sqrt(-I * x - 1), x, 0, '-') == I
assert limit(cbrt(I * x - 1), x, 0) == (-1)**(S(1) / 3)
assert limit(cbrt(I * x - 1), x, 0, '-') == -(-1)**(S(2) / 3)
assert limit(cbrt(-I * x - 1), x, 0) == -(-1)**(S(2) / 3)
assert limit(cbrt(-I * x - 1), x, 0, '-') == (-1)**(S(1) / 3)
def test_conv7b():
x = sympy.Symbol("x")
y = sympy.Symbol("y")
assert sympify(sympy.sin(x/3)) == sin(Symbol("x") / 3)
assert sympify(sympy.sin(x/3)) != cos(Symbol("x") / 3)
assert sympify(sympy.cos(x/3)) == cos(Symbol("x") / 3)
assert sympify(sympy.tan(x/3)) == tan(Symbol("x") / 3)
assert sympify(sympy.cot(x/3)) == cot(Symbol("x") / 3)
assert sympify(sympy.csc(x/3)) == csc(Symbol("x") / 3)
assert sympify(sympy.sec(x/3)) == sec(Symbol("x") / 3)
assert sympify(sympy.asin(x/3)) == asin(Symbol("x") / 3)
assert sympify(sympy.acos(x/3)) == acos(Symbol("x") / 3)
assert sympify(sympy.atan(x/3)) == atan(Symbol("x") / 3)
assert sympify(sympy.acot(x/3)) == acot(Symbol("x") / 3)
assert sympify(sympy.acsc(x/3)) == acsc(Symbol("x") / 3)
assert sympify(sympy.asec(x/3)) == asec(Symbol("x") / 3)
def test_conv7b():
x = sympy.Symbol("x")
y = sympy.Symbol("y")
assert sympify(sympy.sin(x / 3)) == sin(Symbol("x") / 3)
assert sympify(sympy.sin(x / 3)) != cos(Symbol("x") / 3)
assert sympify(sympy.cos(x / 3)) == cos(Symbol("x") / 3)
assert sympify(sympy.tan(x / 3)) == tan(Symbol("x") / 3)
assert sympify(sympy.cot(x / 3)) == cot(Symbol("x") / 3)
assert sympify(sympy.csc(x / 3)) == csc(Symbol("x") / 3)
assert sympify(sympy.sec(x / 3)) == sec(Symbol("x") / 3)
assert sympify(sympy.asin(x / 3)) == asin(Symbol("x") / 3)
assert sympify(sympy.acos(x / 3)) == acos(Symbol("x") / 3)
assert sympify(sympy.atan(x / 3)) == atan(Symbol("x") / 3)
assert sympify(sympy.acot(x / 3)) == acot(Symbol("x") / 3)
assert sympify(sympy.acsc(x / 3)) == acsc(Symbol("x") / 3)
assert sympify(sympy.asec(x / 3)) == asec(Symbol("x") / 3)
def test_asec():
assert asec(nan) == 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(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).as_leading_term(x) == 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
def test_asec():
assert asec(nan) == 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(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).as_leading_term(x) == 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
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) == zoo
assert csc(pi) == zoo
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(5*pi/2) == 1
assert csc(9*pi/7) == -csc(2*pi/7)
assert csc(3*pi/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(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
assert csc(x).rewrite(sin) == 1/sin(x)
assert csc(x).rewrite(cos) == csc(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(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_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).is_finite == False
assert csc(x).is_finite == None
assert csc(pi/2).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
# Funciones para trigonometría (Angulos en radianes)
sp.sin(num) # Seno
sp.cos(num) # Coseno
sp.tan(num) # tangente
sp.cot(num) # cotangente
sp.sec(num) # secante
sp.csc(num) # cosecante
sp.asin(num) # Arcoseno
sp.acos(num) # Arcocoseno
sp.atan(num) # Arcotangente
sp.atan2(catetoY,
catetoX) # Arcotangente de un triangulo segun los catetos (Angulo)
sp.acot(num) # Arcocotangente
sp.asec(num) # Arcosecante
sp.acsc(num) # Arcocosecante
# Funciones hiperbólicas (Angulos en radianes)
sp.sinh(num) # Seno
sp.cosh(num) # Coseno
sp.tanh(num) # tangente
sp.coth(num) # cotangente
sp.sech(num) # secante
sp.csch(num) # cosecante
sp.asinh(num) # Arcoseno
sp.acosh(num) # Arcocoseno
sp.atanh(num) # Arcotangente
sp.acoth(num) # Arcocotangente
sp.asech(num) # Arcosecante
sp.acsch(num) # Arcocosecante
def test_invert_real():
x = Symbol('x', real=True)
y = Symbol('y')
n = Symbol('n')
def ireal(x, s=S.Reals):
return Intersection(s, x)
minus_n = Intersection(Interval(-oo, 0), FiniteSet(-n))
plus_n = Intersection(Interval(0, oo), FiniteSet(n))
assert solveset(abs(x) - n, x, S.Reals) == Union(minus_n, plus_n)
assert invert_real(exp(x), y, x) == (x, ireal(FiniteSet(log(y))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x * 3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3 * x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x) * 3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3 * x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
minus_y = Intersection(Interval(-oo, 0), FiniteSet(-y))
plus_y = Intersection(Interval(0, oo), FiniteSet(y))
assert invert_real(Abs(x), y, x) == (x, Union(minus_y, plus_y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y) / log(2)))
assert invert_real(2**exp(x), y,
x) == (x, ireal(FiniteSet(log(log(y) / log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
y_1 = Intersection(Interval(-1, oo), FiniteSet(y - 1))
y_2 = Intersection(Interval(-oo, -1), FiniteSet(-y - 1))
assert invert_real(Abs(x**31 + x + 1), y,
x) == (x**31 + x, Union(y_1, y_2))
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1 / pi)))
# Test for ``set_h`` containing information about the domain
n = Dummy('n')
x = Symbol('x')
h1 = Intersection(Interval(-oo, -3), FiniteSet(-a + b - 3),
imageset(Lambda(n, n - a - 3), Interval(0, oo)))
h2 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3),
imageset(Lambda(n, -n + a - 3), Interval(0, oo)))
assert invert_real(Abs(Abs(x + 3) - a) - b, 0, x) == (x, Union(h1, h2))
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) == zoo
assert csc(pi) == zoo
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(5 * pi / 2) == 1
assert csc(9 * pi / 7) == -csc(2 * pi / 7)
assert csc(3 * pi / 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(x).rewrite(exp) == 2 * I / (exp(I * x) - exp(-I * x))
assert csc(x).rewrite(sin) == 1 / sin(x)
assert csc(x).rewrite(cos) == csc(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(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_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).is_finite == False
assert csc(x).is_finite == None
assert csc(pi / 2).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
def test_invert_real():
x = Dummy(real=True)
n = Symbol('n')
minus_n = Intersection(Interval(-oo, 0), FiniteSet(-n))
plus_n = Intersection(Interval(0, oo), FiniteSet(n))
assert solveset(abs(x) - n, x, S.Reals) == Union(minus_n, plus_n)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, FiniteSet(log(y - 3)))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
minus_y = Intersection(Interval(-oo, 0), FiniteSet(-y))
plus_y = Intersection(Interval(0, oo), FiniteSet(y))
assert invert_real(Abs(x), y, x) == (x, Union(minus_y, plus_y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, FiniteSet(log(log(y)/log(2))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**Rational(1, 2), y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
raises(ValueError, lambda: invert_real(x**pi, y, x))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
y_1 = Intersection(Interval(-1, oo), FiniteSet(y - 1))
y_2 = Intersection(Interval(-oo, -1), FiniteSet(-y - 1))
assert invert_real(Abs(x**31 + x + 1), y, x) == (x**31 + x,
Union(y_1, y_2))
assert invert_real(sin(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers))
assert invert_real(sin(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers))
assert invert_real(csc(x), y, x) == \
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers))
assert invert_real(csc(exp(x)), y, x) == \
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers))
assert invert_real(cos(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers)))
assert invert_real(cos(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers)))
assert invert_real(sec(x), y, x) == \
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers)))
assert invert_real(sec(exp(x)), y, x) == \
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers)))
assert invert_real(tan(x), y, x) == \
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers))
assert invert_real(tan(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers))
assert invert_real(cot(x), y, x) == \
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers))
assert invert_real(cot(exp(x)), y, x) == \
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers))
assert invert_real(tan(tan(x)), y, x) == \
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
# Test for ``set_h`` containing information about the domain
n = Dummy('n')
x = Symbol('x')
h1 = Intersection(Interval(-3, oo), FiniteSet(a + b - 3),
imageset(Lambda(n, -n + a - 3), Interval(-oo, 0)))
h2 = Intersection(Interval(-oo, -3), FiniteSet(-a + b - 3),
imageset(Lambda(n, n - a - 3), Interval(0, oo)))
h3 = Intersection(Interval(-3, oo), FiniteSet(a - b - 3),
imageset(Lambda(n, -n + a - 3), Interval(0, oo)))
h4 = Intersection(Interval(-oo, -3), FiniteSet(-a - b - 3),
imageset(Lambda(n, n - a - 3), Interval(-oo, 0)))
assert invert_real(Abs(Abs(x + 3) - a) - b, 0, x) == (x, Union(h1, h2, h3, h4))
def test_manualintegrate_inversetrig():
# atan
assert manualintegrate(exp(x) / (1 + exp(2 * x)), x) == atan(exp(x))
assert manualintegrate(1 / (4 + 9 * x**2), x) == atan(3 * x / 2) / 6
assert manualintegrate(1 / (16 + 16 * x**2), x) == atan(x) / 16
assert manualintegrate(1 / (4 + x**2), x) == atan(x / 2) / 2
assert manualintegrate(1 / (1 + 4 * x**2), x) == atan(2 * x) / 2
ra = Symbol('a', real=True)
rb = Symbol('b', real=True)
assert manualintegrate(1/(ra + rb*x**2), x) == \
Piecewise((atan(x/sqrt(ra/rb))/(rb*sqrt(ra/rb)), ra/rb > 0),
(-acoth(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 > -ra/rb)),
(-atanh(x/sqrt(-ra/rb))/(rb*sqrt(-ra/rb)), And(ra/rb < 0, x**2 < -ra/rb)))
assert manualintegrate(1/(4 + rb*x**2), x) == \
Piecewise((atan(x/(2*sqrt(1/rb)))/(2*rb*sqrt(1/rb)), 4/rb > 0),
(-acoth(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 > -4/rb)),
(-atanh(x/(2*sqrt(-1/rb)))/(2*rb*sqrt(-1/rb)), And(4/rb < 0, x**2 < -4/rb)))
assert manualintegrate(1/(ra + 4*x**2), x) == \
Piecewise((atan(2*x/sqrt(ra))/(2*sqrt(ra)), ra/4 > 0),
(-acoth(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 > -ra/4)),
(-atanh(2*x/sqrt(-ra))/(2*sqrt(-ra)), And(ra/4 < 0, x**2 < -ra/4)))
assert manualintegrate(1 / (4 + 4 * x**2), x) == atan(x) / 4
assert manualintegrate(1 / (a + b * x**2),
x) == atan(x / sqrt(a / b)) / (b * sqrt(a / b))
# asin
assert manualintegrate(1 / sqrt(1 - x**2), x) == asin(x)
assert manualintegrate(1 / sqrt(4 - 4 * x**2), x) == asin(x) / 2
assert manualintegrate(3 / sqrt(1 - 9 * x**2), x) == asin(3 * x)
assert manualintegrate(1 / sqrt(4 - 9 * x**2),
x) == asin(x * Rational(3, 2)) / 3
# asinh
assert manualintegrate(1/sqrt(x**2 + 1), x) == \
asinh(x)
assert manualintegrate(1/sqrt(x**2 + 4), x) == \
asinh(x/2)
assert manualintegrate(1/sqrt(4*x**2 + 4), x) == \
asinh(x)/2
assert manualintegrate(1/sqrt(4*x**2 + 1), x) == \
asinh(2*x)/2
assert manualintegrate(1/sqrt(a*x**2 + 1), x) == \
Piecewise((sqrt(-1/a)*asin(x*sqrt(-a)), a < 0), (sqrt(1/a)*asinh(sqrt(a)*x), a > 0))
assert manualintegrate(1/sqrt(a + x**2), x) == \
Piecewise((asinh(x*sqrt(1/a)), a > 0), (acosh(x*sqrt(-1/a)), a < 0))
# acosh
assert manualintegrate(1/sqrt(x**2 - 1), x) == \
acosh(x)
assert manualintegrate(1/sqrt(x**2 - 4), x) == \
acosh(x/2)
assert manualintegrate(1/sqrt(4*x**2 - 4), x) == \
acosh(x)/2
assert manualintegrate(1/sqrt(9*x**2 - 1), x) == \
acosh(3*x)/3
assert manualintegrate(1/sqrt(a*x**2 - 4), x) == \
Piecewise((sqrt(1/a)*acosh(sqrt(a)*x/2), a > 0))
assert manualintegrate(1/sqrt(-a + 4*x**2), x) == \
Piecewise((asinh(2*x*sqrt(-1/a))/2, -a > 0), (acosh(2*x*sqrt(1/a))/2, -a < 0))
# From https://www.wikiwand.com/en/List_of_integrals_of_inverse_trigonometric_functions
# asin
assert manualintegrate(asin(x), x) == x * asin(x) + sqrt(1 - x**2)
assert manualintegrate(asin(a * x), x) == Piecewise(
((a * x * asin(a * x) + sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)),
(0, True))
assert manualintegrate(x * asin(a * x), x) == -a * Integral(
x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * asin(a * x) / 2
# acos
assert manualintegrate(acos(x), x) == x * acos(x) - sqrt(1 - x**2)
assert manualintegrate(acos(a * x), x) == Piecewise(
((a * x * acos(a * x) - sqrt(-a**2 * x**2 + 1)) / a, Ne(a, 0)),
(pi * x / 2, True))
assert manualintegrate(x * acos(a * x), x) == a * Integral(
x**2 / sqrt(-a**2 * x**2 + 1), x) / 2 + x**2 * acos(a * x) / 2
# atan
assert manualintegrate(atan(x), x) == x * atan(x) - log(x**2 + 1) / 2
assert manualintegrate(atan(a * x), x) == Piecewise(
((a * x * atan(a * x) - log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)),
(0, True))
assert manualintegrate(
x * atan(a * x),
x) == -a * (x / a**2 - atan(x / sqrt(a**(-2))) /
(a**4 * sqrt(a**(-2)))) / 2 + x**2 * atan(a * x) / 2
# acsc
assert manualintegrate(
acsc(x), x) == x * acsc(x) + Integral(1 / (x * sqrt(1 - 1 / x**2)), x)
assert manualintegrate(
acsc(a * x),
x) == x * acsc(a * x) + Integral(1 / (x * sqrt(1 - 1 /
(a**2 * x**2))), x) / a
assert manualintegrate(x * acsc(a * x),
x) == x**2 * acsc(a * x) / 2 + Integral(
1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a)
# asec
assert manualintegrate(
asec(x), x) == x * asec(x) - Integral(1 / (x * sqrt(1 - 1 / x**2)), x)
assert manualintegrate(
asec(a * x),
x) == x * asec(a * x) - Integral(1 / (x * sqrt(1 - 1 /
(a**2 * x**2))), x) / a
assert manualintegrate(x * asec(a * x),
x) == x**2 * asec(a * x) / 2 - Integral(
1 / sqrt(1 - 1 / (a**2 * x**2)), x) / (2 * a)
# acot
assert manualintegrate(acot(x), x) == x * acot(x) + log(x**2 + 1) / 2
assert manualintegrate(acot(a * x), x) == Piecewise(
((a * x * acot(a * x) + log(a**2 * x**2 + 1) / 2) / a, Ne(a, 0)),
(pi * x / 2, True))
assert manualintegrate(
x * acot(a * x),
x) == a * (x / a**2 - atan(x / sqrt(a**(-2))) /
(a**4 * sqrt(a**(-2)))) / 2 + x**2 * acot(a * x) / 2
# piecewise
assert manualintegrate(1/sqrt(a-b*x**2), x) == \
Piecewise((sqrt(a/b)*asin(x*sqrt(b/a))/sqrt(a), And(-b < 0, a > 0)),
(sqrt(-a/b)*asinh(x*sqrt(-b/a))/sqrt(a), And(-b > 0, a > 0)),
(sqrt(a/b)*acosh(x*sqrt(b/a))/sqrt(-a), And(-b > 0, a < 0)))
assert manualintegrate(1/sqrt(a + b*x**2), x) == \
Piecewise((sqrt(-a/b)*asin(x*sqrt(-b/a))/sqrt(a), And(a > 0, b < 0)),
(sqrt(a/b)*asinh(x*sqrt(b/a))/sqrt(a), And(a > 0, b > 0)),
(sqrt(-a/b)*acosh(x*sqrt(-b/a))/sqrt(-a), And(a < 0, b > 0)))
