Has anybody an explanation for this output?
print bool( tan(pi/6) == sin(pi/6)/cos(pi/6) )
print bool( tan(pi/5) == sin(pi/5)/cos(pi/5) )
print bool( tan(pi/5).n() == sin(pi/5).n()/cos(pi/5).n() )
True
False
True
Looks like a bug to me. I do get True for the following.
eq=tan(pi/5) == sin(pi/5)/cos(pi/5)
bool(eq.simplify_trig())
Note that eq.simplify_trig() gives:
4*sin(1/5*pi)/(sqrt(5) + 1) == 4*sin(1/5*pi)/(sqrt(5) + 1)
On the same line, one can check that things like sin((n-1)*pi/n) == sin(pi/n) give False for a general n, and there simplify_trig does not help :(. The origin is different, anyway: probably Sage should send every angle inside a trig function to the first quadrant, as Mathematica does: e.g. sin((n-1)*pi/n) returns sin(pi/n), for a general n.
This problem is likely related to the one I reported at: http://ask.sagemath.org/question/1850/odd-trig-function-behavior
Sage (through Pynac and GiNaC) automatically evaluates sin(pi/6), cos(pi/6), tan(pi/6):
sage: sin(pi/6)
1/2
sage: cos(pi/6)
1/2*sqrt(3)
sage: tan(pi/6)
1/3*sqrt(3)
For the comparison, it checks if tan(pi/6) - sin(pi/6)/cos(pi/6) is 0:
sage: t = tan(pi/6) - sin(pi/6)/cos(pi/6)
sage: t
0
sage: t.is_trivial_zero()
True
For pi/5, tan() and sin() are left unevaluated, while cos() is:
sage: tan(pi/5)
tan(1/5*pi)
sage: sin(pi/5)
sin(1/5*pi)
sage: cos(pi/5)
1/4*sqrt(5) + 1/4
This makes the difference tan(pi/5) - sin(pi/5)/cos(pi/5) a symbolic expression not trivially equal to 0:
sage: t = tan(pi/5) - sin(pi/5)/cos(pi/5)
sage: t
-4*sin(1/5*pi)/(sqrt(5) + 1) + tan(1/5*pi)
sage: t.is_trivial_zero()
False
sage: t.n()
1.11022302462516e-16
Any patches to improve this behavior would be more than welcome.
Asked: 2012-11-08 17:43:39 +0100
Seen: 831 times
Last updated: Nov 10 '12
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.