Trigonometric Functions¶
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class
sage.functions.trig.
Function_arccos
¶ Bases:
sage.symbolic.function.GinacFunction
The arccosine function.
EXAMPLES:
sage: arccos(0.5) 1.04719755119660 sage: arccos(1/2) 1/3*pi sage: arccos(1 + 1.0*I) 0.904556894302381 - 1.06127506190504*I sage: arccos(3/4).n(100) 0.72273424781341561117837735264
We can delay evaluation using the
hold
parameter:sage: arccos(0,hold=True) arccos(0)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arccos(0,hold=True); a.simplify() 1/2*pi
conjugate(arccos(x))==arccos(conjugate(x))
, unless on the branch cuts, which run along the real axis outside the interval [-1, +1].:sage: conjugate(arccos(x)) conjugate(arccos(x)) sage: var('y', domain='positive') y sage: conjugate(arccos(y)) conjugate(arccos(y)) sage: conjugate(arccos(y+I)) conjugate(arccos(y + I)) sage: conjugate(arccos(1/16)) arccos(1/16) sage: conjugate(arccos(2)) conjugate(arccos(2)) sage: conjugate(arccos(-2)) pi - conjugate(arccos(2))
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class
sage.functions.trig.
Function_arccot
¶ Bases:
sage.symbolic.function.GinacFunction
The arccotangent function.
EXAMPLES:
sage: arccot(1/2) arccot(1/2) sage: RDF(arccot(1/2)) # abs tol 2e-16 1.1071487177940906 sage: arccot(1 + I) arccot(I + 1) sage: arccot(1/2).n(100) 1.1071487177940905030170654602 sage: float(arccot(1/2)) # abs tol 2e-16 1.1071487177940906 sage: bool(diff(acot(x), x) == -diff(atan(x), x)) True sage: diff(acot(x), x) -1/(x^2 + 1)
We can delay evaluation using the
hold
parameter:sage: arccot(1,hold=True) arccot(1)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arccot(1,hold=True); a.simplify() 1/4*pi
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class
sage.functions.trig.
Function_arccsc
¶ Bases:
sage.symbolic.function.GinacFunction
The arccosecant function.
EXAMPLES:
sage: arccsc(2) arccsc(2) sage: RDF(arccsc(2)) # rel tol 1e-15 0.5235987755982988 sage: arccsc(2).n(100) 0.52359877559829887307710723055 sage: float(arccsc(2)) 0.52359877559829... sage: arccsc(1 + I) arccsc(I + 1) sage: diff(acsc(x), x) -1/(sqrt(x^2 - 1)*x) sage: arccsc(x)._sympy_() acsc(x)
We can delay evaluation using the
hold
parameter:sage: arccsc(1,hold=True) arccsc(1)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arccsc(1,hold=True); a.simplify() 1/2*pi
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class
sage.functions.trig.
Function_arcsec
¶ Bases:
sage.symbolic.function.GinacFunction
The arcsecant function.
EXAMPLES:
sage: arcsec(2) arcsec(2) sage: arcsec(2.0) 1.04719755119660 sage: arcsec(2).n(100) 1.0471975511965977461542144611 sage: arcsec(1/2).n(100) 1.3169578969248167086250463473*I sage: RDF(arcsec(2)) # abs tol 1e-15 1.0471975511965976 sage: arcsec(1 + I) arcsec(I + 1) sage: diff(asec(x), x) 1/(sqrt(x^2 - 1)*x) sage: arcsec(x)._sympy_() asec(x)
We can delay evaluation using the
hold
parameter:sage: arcsec(1,hold=True) arcsec(1)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arcsec(1,hold=True); a.simplify() 0
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class
sage.functions.trig.
Function_arcsin
¶ Bases:
sage.symbolic.function.GinacFunction
The arcsine function.
EXAMPLES:
sage: arcsin(0.5) 0.523598775598299 sage: arcsin(1/2) 1/6*pi sage: arcsin(1 + 1.0*I) 0.666239432492515 + 1.06127506190504*I
We can delay evaluation using the
hold
parameter:sage: arcsin(0,hold=True) arcsin(0)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arcsin(0,hold=True); a.simplify() 0
conjugate(arcsin(x))==arcsin(conjugate(x))
, unless on the branch cuts which run along the real axis outside the interval [-1, +1].:sage: conjugate(arcsin(x)) conjugate(arcsin(x)) sage: var('y', domain='positive') y sage: conjugate(arcsin(y)) conjugate(arcsin(y)) sage: conjugate(arcsin(y+I)) conjugate(arcsin(y + I)) sage: conjugate(arcsin(1/16)) arcsin(1/16) sage: conjugate(arcsin(2)) conjugate(arcsin(2)) sage: conjugate(arcsin(-2)) -conjugate(arcsin(2))
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class
sage.functions.trig.
Function_arctan
¶ Bases:
sage.symbolic.function.GinacFunction
The arctangent function.
EXAMPLES:
sage: arctan(1/2) arctan(1/2) sage: RDF(arctan(1/2)) # rel tol 1e-15 0.46364760900080615 sage: arctan(1 + I) arctan(I + 1) sage: arctan(1/2).n(100) 0.46364760900080611621425623146
We can delay evaluation using the
hold
parameter:sage: arctan(0,hold=True) arctan(0)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = arctan(0,hold=True); a.simplify() 0
conjugate(arctan(x))==arctan(conjugate(x))
, unless on the branch cuts which run along the imaginary axis outside the interval [-I, +I].:sage: conjugate(arctan(x)) conjugate(arctan(x)) sage: var('y', domain='positive') y sage: conjugate(arctan(y)) arctan(y) sage: conjugate(arctan(y+I)) conjugate(arctan(y + I)) sage: conjugate(arctan(1/16)) arctan(1/16) sage: conjugate(arctan(-2*I)) conjugate(arctan(-2*I)) sage: conjugate(arctan(2*I)) conjugate(arctan(2*I)) sage: conjugate(arctan(I/2)) arctan(-1/2*I)
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class
sage.functions.trig.
Function_arctan2
¶ Bases:
sage.symbolic.function.GinacFunction
The modified arctangent function.
Returns the arc tangent (measured in radians) of \(y/x\), where unlike
arctan(y/x)
, the signs of bothx
andy
are considered. In particular, this function measures the angle of a ray through the origin and \((x,y)\), with the positive \(x\)-axis the zero mark, and with output angle \(\theta\) being between \(-\pi<\theta<=\pi\).Hence,
arctan2(y,x) = arctan(y/x)
only for \(x>0\). One may consider the usual arctan to measure angles of lines through the origin, while the modified function measures rays through the origin.Note that the \(y\)-coordinate is by convention the first input.
EXAMPLES:
Note the difference between the two functions:
sage: arctan2(1,-1) 3/4*pi sage: arctan(1/-1) -1/4*pi
This is consistent with Python and Maxima:
sage: maxima.atan2(1,-1) (3*%pi)/4 sage: math.atan2(1,-1) 2.356194490192345
More examples:
sage: arctan2(1,0) 1/2*pi sage: arctan2(2,3) arctan(2/3) sage: arctan2(-1,-1) -3/4*pi
Of course we can approximate as well:
sage: arctan2(-1/2,1).n(100) -0.46364760900080611621425623146 sage: arctan2(2,3).n(100) 0.58800260354756755124561108063
We can delay evaluation using the
hold
parameter:sage: arctan2(-1/2,1,hold=True) arctan2(-1/2, 1)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: arctan2(-1/2,1,hold=True).simplify() -arctan(1/2)
The function also works with numpy arrays as input:
sage: import numpy sage: a = numpy.linspace(1, 3, 3) sage: b = numpy.linspace(3, 6, 3) sage: atan2(a, b) array([0.32175055, 0.41822433, 0.46364761]) sage: atan2(1,a) array([0.78539816, 0.46364761, 0.32175055]) sage: atan2(a, 1) array([0.78539816, 1.10714872, 1.24904577])
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class
sage.functions.trig.
Function_cos
¶ Bases:
sage.symbolic.function.GinacFunction
The cosine function.
EXAMPLES:
sage: cos(pi) -1 sage: cos(x).subs(x==pi) -1 sage: cos(2).n(100) -0.41614683654714238699756822950 sage: loads(dumps(cos)) cos sage: cos(x)._sympy_() cos(x)
We can prevent evaluation using the
hold
parameter:sage: cos(0,hold=True) cos(0)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = cos(0,hold=True); a.simplify() 1
If possible, the argument is also reduced modulo the period length \(2\pi\), and well-known identities are directly evaluated:
sage: k = var('k', domain='integer') sage: cos(1 + 2*k*pi) cos(1) sage: cos(k*pi) cos(pi*k) sage: cos(pi/3 + 2*k*pi) 1/2
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class
sage.functions.trig.
Function_cot
¶ Bases:
sage.symbolic.function.GinacFunction
The cotangent function.
EXAMPLES:
sage: cot(pi/4) 1 sage: RR(cot(pi/4)) 1.00000000000000 sage: cot(1/2) cot(1/2) sage: cot(0.5) 1.83048772171245 sage: latex(cot(x)) \cot\left(x\right) sage: cot(x)._sympy_() cot(x)
We can prevent evaluation using the
hold
parameter:sage: cot(pi/4,hold=True) cot(1/4*pi)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = cot(pi/4,hold=True); a.simplify() 1
EXAMPLES:
sage: cot(pi/4) 1 sage: cot(x).subs(x==pi/4) 1 sage: cot(pi/7) cot(1/7*pi) sage: cot(x) cot(x) sage: n(cot(pi/4),100) 1.0000000000000000000000000000 sage: float(cot(1)) 0.64209261593433... sage: bool(diff(cot(x), x) == diff(1/tan(x), x)) True sage: diff(cot(x), x) -cot(x)^2 - 1
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class
sage.functions.trig.
Function_csc
¶ Bases:
sage.symbolic.function.GinacFunction
The cosecant function.
EXAMPLES:
sage: csc(pi/4) sqrt(2) sage: csc(x).subs(x==pi/4) sqrt(2) sage: csc(pi/7) csc(1/7*pi) sage: csc(x) csc(x) sage: RR(csc(pi/4)) 1.41421356237310 sage: n(csc(pi/4),100) 1.4142135623730950488016887242 sage: float(csc(pi/4)) 1.4142135623730951 sage: csc(1/2) csc(1/2) sage: csc(0.5) 2.08582964293349 sage: bool(diff(csc(x), x) == diff(1/sin(x), x)) True sage: diff(csc(x), x) -cot(x)*csc(x) sage: latex(csc(x)) \csc\left(x\right) sage: csc(x)._sympy_() csc(x)
We can prevent evaluation using the
hold
parameter:sage: csc(pi/4,hold=True) csc(1/4*pi)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = csc(pi/4,hold=True); a.simplify() sqrt(2)
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class
sage.functions.trig.
Function_sec
¶ Bases:
sage.symbolic.function.GinacFunction
The secant function.
EXAMPLES:
sage: sec(pi/4) sqrt(2) sage: sec(x).subs(x==pi/4) sqrt(2) sage: sec(pi/7) sec(1/7*pi) sage: sec(x) sec(x) sage: RR(sec(pi/4)) 1.41421356237310 sage: n(sec(pi/4),100) 1.4142135623730950488016887242 sage: float(sec(pi/4)) 1.4142135623730951 sage: sec(1/2) sec(1/2) sage: sec(0.5) 1.13949392732455 sage: bool(diff(sec(x), x) == diff(1/cos(x), x)) True sage: diff(sec(x), x) sec(x)*tan(x) sage: latex(sec(x)) \sec\left(x\right) sage: sec(x)._sympy_() sec(x)
We can prevent evaluation using the
hold
parameter:sage: sec(pi/4,hold=True) sec(1/4*pi)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = sec(pi/4,hold=True); a.simplify() sqrt(2)
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class
sage.functions.trig.
Function_sin
¶ Bases:
sage.symbolic.function.GinacFunction
The sine function.
EXAMPLES:
sage: sin(0) 0 sage: sin(x).subs(x==0) 0 sage: sin(2).n(100) 0.90929742682568169539601986591 sage: loads(dumps(sin)) sin sage: sin(x)._sympy_() sin(x)
We can prevent evaluation using the
hold
parameter:sage: sin(0,hold=True) sin(0)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = sin(0,hold=True); a.simplify() 0
If possible, the argument is also reduced modulo the period length \(2\pi\), and well-known identities are directly evaluated:
sage: k = var('k', domain='integer') sage: sin(1 + 2*k*pi) sin(1) sage: sin(k*pi) 0
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class
sage.functions.trig.
Function_tan
¶ Bases:
sage.symbolic.function.GinacFunction
The tangent function.
EXAMPLES:
sage: tan(pi) 0 sage: tan(3.1415) -0.0000926535900581913 sage: tan(3.1415/4) 0.999953674278156 sage: tan(pi/4) 1 sage: tan(1/2) tan(1/2) sage: RR(tan(1/2)) 0.546302489843790
We can prevent evaluation using the
hold
parameter:sage: tan(pi/4,hold=True) tan(1/4*pi)
To then evaluate again, we currently must use Maxima via
sage.symbolic.expression.Expression.simplify()
:sage: a = tan(pi/4,hold=True); a.simplify() 1
If possible, the argument is also reduced modulo the period length \(\pi\), and well-known identities are directly evaluated:
sage: k = var('k', domain='integer') sage: tan(1 + 2*k*pi) tan(1) sage: tan(k*pi) 0