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Trigonometric equality (with arctan and arcsin)

asked 2017-02-23 15:29:32 +0200

Romuald_314 gravatar image

updated 2017-02-23 15:35:42 +0200


I found a strange issue:

ex = (arctan(1/2) + arcsin(sqrt(5)/5) == arctan(4/3))

And it gives False.

A simple numeric approximation allows to see that it's true, and wolfram alpha for instance gives the correct answer. But why is it like that and how can I fix it?


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In sagecell, giac('simplify(tExpand((arctan(1/2) + arcsin(sqrt(5)/5) - arctan(4/3))))') returns $0$. tExpand is a shortcut for expand trascendental expressions.

mforets gravatar imagemforets ( 2017-02-23 23:24:57 +0200 )edit

Seems to work. Thanks for the terminology.

Romuald_314 gravatar imageRomuald_314 ( 2017-02-27 17:38:03 +0200 )edit

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answered 2017-02-23 16:40:48 +0200

eric_g gravatar image

updated 2017-02-23 16:43:20 +0200

Actually, Sage returns False whenever it does not manage to prove that an equality is true or false. Therefore only True is a mathematically meaningful output for such a test.

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Oh, that's problematical. It's a shame not to be able to verify such an equality. Really, there is no way?

Romuald_314 gravatar imageRomuald_314 ( 2017-02-23 18:51:40 +0200 )edit

answered 2017-02-25 17:51:44 +0200

dan_fulea gravatar image

I tried:

sage: tan( arctan(1/2) + arcsin(sqrt(5)/5) - arctan(4/3) ).simplify_full()


Usually, it is a good idea to put all expressions on one side. The symbolic expression simplifier has a simpler work.

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Your way is also working, however I find weird having to add a 'tan(...)' to the expression to prove that it's equal to 0. Indeed, tan(0)=0 but that does not always work (tan(2k*pi)=0 too)

Romuald_314 gravatar imageRomuald_314 ( 2017-02-27 17:44:01 +0200 )edit

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Asked: 2017-02-23 15:29:32 +0200

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Last updated: Feb 25 '17