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Re: [Axiom-developer] unexpected behaviour of normalize(1-(cos(x))^2)
From: |
Bill Page |
Subject: |
Re: [Axiom-developer] unexpected behaviour of normalize(1-(cos(x))^2) |
Date: |
Thu, 13 Aug 2009 00:57:22 -0400 |
Why not use removeCosSq ?
(3) -> removeCosSq(f)
2
(3) sin(x)
Type: Expression(Integer)
(4) -> removeCosSq(f+3)
2
(4) sin(x) + 3
Type: Expression(Integer)
(5) -> removeCosSq(f+a)
2
(5) sin(x) + a
Type: Expression(Integer)
(6) -> removeCosSq(2*(f+a))
2
(6) 2sin(x) + 2a
Type: Expression(Integer)
(7) -> removeCosSq(1/(f+a))
1
(7) -----------
2
sin(x) + a
Type: Expression(Integer)
See also:
removeCoshSq
removeSinSq
removeSinhSq
Regards,
Bill Page.
On Wed, Aug 12, 2009 at 10:47 PM, Michael Becker wrote:
> Am Mittwoch, 5. August 2009 15:59 schrieben Sie:
>> Michael,
>>
>> Trig identity substitutions are somewhat problematic in Axiom.
>> See the src/input/schaum* files for examples.
>>
>> If the subexpression (1-cos(x)^2) occurs in your expression E you can
>> write:
>>
>> sinrule:=rule((1-cos(x)^2) == sin(x)^2)
>>
>> and then use this rule for your expression E thus
>>
>> sinrule(E)
>
>
>
>
> Tim,
>
>
> this does not always work (see (6) and (7)) :
>
>
>
> (1) -> )set mess auto off
> (1) -> sinrule:=rule((1-cos(x)^2) == sin(x)^2)
> (1) ->
> 2 2
> (1) - cos(x) + %C + 1 == sin(x) + %C
> Type: RewriteRule(Integer,Integer,Expression Integer)
> (2) -> f:= 1 - cos(x)^2
> (2) ->
> 2
> (2) - cos(x) + 1
> Type: Expression Integer
> (3) -> sinrule(f)
> (3) ->
> 2
> (3) sin(x)
> Type: Expression Integer
> (4) -> sinrule(f+3)
> (4) ->
> 2
> (4) - cos(x) + 4
> Type: Expression Integer
> (5) -> sinrule(f+a)
> (5) ->
> 2
> (5) sin(x) + a
> Type: Expression Integer
> (6) -> sinrule (2*(f+a))
> (6) ->
> 2
> (6) - 2cos(x) + 2a + 2
> Type: Expression Integer
> (7) -> sinrule (1/(f+a))
> (7) ->
> 1
> (7) - ---------------
> 2
> cos(x) - a - 1
> Type: Expression Integer
>
>
>
>
> - Michael
>
>
>
>
>
>>
>> Axiom will not derive several of the trig identities from scratch.
>>
>> In your expression we have something of the form
>> (4a^2) / (a^2 + 1)^2 where a = tan(x/2)
>> so Axiom needs to show that
>> (a^2+1)^2 != 0
>> (a^2+1) != 0
>> a^2 != -1
>> a != i
>> or, by back-substitution
>> tan(x/2) != i
>> which it does not conclude automatically, even though this
>> is clearly true in the domain Expression(Integer).
>>
>> Michael Becker wrote:
>> > Hi,
>> >
>> >
>> > Is this (30) the expected bevaviour of 'normalize' ??
>> >
>> >
>> > (29) -> normalize ((sin(x))^2+(cos(x))^2)
>> > (29) ->
>> > (29) 1
>> > Type: Expression
>> > Integer
>> >
>> >
>> >
>> > (30) -> normalize (1-(cos(x))^2)
>> > (30) ->
>> > x 2
>> > 4tan(-)
>> > 2
>> > (30) ----------------------
>> > x 4 x 2
>> > tan(-) + 2tan(-) + 1
>> > 2 2
>> > Type: Expression
>> > Integer
>> >
>> >
>> >
>> >
>> >
>> >
>> > -- Michael
>
>
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