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))
TESTS:
sage: arccos(x)._sympy_() acos(x) sage: arccos(x).operator() arccos sage: acos(complex(1,1)) (0.9045568943023814-1.0612750619050357j)
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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
TESTS:
Test complex input:
sage: arccot(x)._sympy_() acot(x) sage: arccot(complex(1,1)) # rel tol 1e-15 (0.5535743588970452-0.4023594781085251j) sage: arccot(1.+I) 0.553574358897045 - 0.402359478108525*I
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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
TESTS:
Test complex input:
sage: arccsc(complex(1,1)) # rel tol 1e-15 (0.45227844715119064-0.5306375309525178j)
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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) NaN 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
TESTS:
Test complex input:
sage: arcsec(complex(1,1)) # rel tol 1e-15 (1.118517879643706+0.5306375309525178j)
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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))
TESTS:
sage: arcsin(x)._sympy_() asin(x) sage: arcsin(x).operator() arcsin sage: asin(complex(1,1)) (0.6662394324925152+1.0612750619050357j)
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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)
TESTS:
sage: arctan(x)._sympy_() atan(x) sage: arctan(x).operator() arctan sage: atan(complex(1,1)) (1.0172219678978514+0.4023594781085251j)
Check that trac ticket #19918 is fixed:
sage: arctan(-x).subs(x=oo) -1/2*pi sage: arctan(-x).subs(x=-oo) 1/2*pi
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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])
TESTS:
sage: x,y = var('x,y') sage: arctan2(y,x).operator() arctan2
Check if trac ticket #8565 is fixed:
sage: atan2(-pi,0) -1/2*pi
Check if trac ticket #8564 is fixed:
sage: arctan2(x,x)._sympy_() atan2(x, x)
Check if numerical evaluation works trac ticket #9913:
sage: arctan2(0, -log(2)).n() 3.14159265358979
Check if atan2(0,0) throws error of trac ticket #11423:
sage: atan2(0,0) NaN sage: atan2(0,0,hold=True) arctan2(0, 0) sage: atan2(0,0,hold=True).n() Traceback (most recent call last): ... ValueError: arctan2(0,0) undefined
Check if trac ticket #10062 is fixed, this was caused by
(I*I).is_positive()
returningTrue
:sage: arctan2(0, I*I) pi
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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
TESTS:
sage: conjugate(cos(x)) cos(conjugate(x)) sage: cos(complex(1,1)) # rel tol 1e-15 (0.8337300251311491-0.9888977057628651j)
Check that trac ticket #20752 is fixed:
sage: cos(3*pi+41/42*pi) cos(1/42*pi) sage: cos(-5*pi+1/42*pi) -cos(1/42*pi) sage: cos(pi-1/42*pi) -cos(1/42*pi)
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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
TESTS:
Test complex input:
sage: cot(complex(1,1)) # rel tol 1e-15 (0.21762156185440273-0.8680141428959249j) sage: cot(1.+I) 0.217621561854403 - 0.868014142895925*I
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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)
TESTS:
Test complex input:
sage: csc(complex(1,1)) # rel tol 1e-15 (0.6215180171704284-0.30393100162842646j)
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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)
TESTS:
Test complex input:
sage: sec(complex(1,1)) # rel tol 1e-15 (0.49833703055518686+0.5910838417210451j)
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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
TESTS:
sage: conjugate(sin(x)) sin(conjugate(x)) sage: sin(complex(1,1)) # rel tol 1e-15 (1.2984575814159773+0.6349639147847361j) sage: sin(pi/5) 1/4*sqrt(-2*sqrt(5) + 10) sage: sin(pi/8) 1/2*sqrt(-sqrt(2) + 2) sage: sin(pi/24) 1/4*sqrt(-2*sqrt(6) - 2*sqrt(2) + 8) sage: sin(pi/30) -1/8*sqrt(5) + 1/4*sqrt(-3/2*sqrt(5) + 15/2) - 1/8 sage: sin(104*pi/105) sin(1/105*pi) sage: cos(pi/8) 1/2*sqrt(sqrt(2) + 2) sage: cos(pi/10) 1/4*sqrt(2*sqrt(5) + 10) sage: cos(pi/12) 1/4*sqrt(6) + 1/4*sqrt(2) sage: cos(pi/15) 1/8*sqrt(5) + 1/4*sqrt(3/2*sqrt(5) + 15/2) - 1/8 sage: cos(pi/24) 1/4*sqrt(2*sqrt(6) + 2*sqrt(2) + 8) sage: cos(104*pi/105) -cos(1/105*pi) sage: tan(pi/5) sqrt(-2*sqrt(5) + 5) sage: tan(pi/8) sqrt(2) - 1 sage: tan(pi/10) 1/5*sqrt(-10*sqrt(5) + 25) sage: tan(pi/16) -sqrt(2) + sqrt(2*sqrt(2) + 4) - 1 sage: tan(pi/20) sqrt(5) - sqrt(2*sqrt(5) + 5) + 1 sage: tan(pi/24) sqrt(6) - sqrt(3) + sqrt(2) - 2 sage: tan(104*pi/105) -tan(1/105*pi) sage: cot(104*pi/105) -cot(1/105*pi) sage: sec(104*pi/105) -sec(1/105*pi) sage: csc(104*pi/105) csc(1/105*pi) sage: all(sin(rat*pi).n(200)-sin(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/5,2/5,1/30,7/30,11/30,13/30,1/8,3/8,1/24,5/24,7/24,11/24]) True sage: all(cos(rat*pi).n(200)-cos(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/10,3/10,1/12,5/12,1/15,2/15,4/15,7/15,1/8,3/8,1/24,5/24,11/24]) True sage: all(tan(rat*pi).n(200)-tan(rat*pi,hold=True).n(200) < 1e-30 for rat in [1/5,2/5,1/10,3/10,1/20,3/20,7/20,9/20,1/8,3/8,1/16,3/16,5/16,7/16,1/24,5/24,7/24,11/24]) True
Check that trac ticket #20456 is fixed:
sage: assume(x>0) sage: sin(pi*x) sin(pi*x) sage: forget()
Check that trac ticket #20752 is fixed:
sage: sin(3*pi+41/42*pi) -sin(1/42*pi) sage: sin(-5*pi+1/42*pi) -sin(1/42*pi) sage: sin(pi-1/42*pi) sin(1/42*pi)
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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
TESTS:
sage: tan(x)._sympy_() tan(x) sage: conjugate(tan(x)) tan(conjugate(x)) sage: tan(complex(1,1)) # rel tol 1e-15 (0.2717525853195118+1.0839233273386946j)
Check that trac ticket #19791 is fixed:
sage: tan(2+I).imag().n() 1.16673625724092