Function of an angle 'A'.
V.Gopal
find/calculate the value of Tangent of an angle without actually
assigning values to perpendicular and base, in which case triangle is
not visualized? That is, if there is a very small gap in the curve
that shows the relation between 'A' (angle) and TanA, can we find the
value of TanB, where B is an angle that lies in the gap? Is Tangent of
an angle 'unpredictable'?
Uncle Al
TanA is a smooth and almost continuous function with countable
singularities in known places. Linearly interpolate. If you want a
better number, use Newtonian approximation.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?" The Net!
Bryan Reed
V.Gopal <vgop...@rediffmail.com> wrote:
>Is there any method - statistical or any other - by which we can
>find/calculate the value of Tangent of an angle without actually
>assigning values to perpendicular and base, in which case triangle is
>not visualized?
Of course. How do you think calculators do it? Ever heard of a Taylor
series?
Bryan
Big Bird
> Is there any method - statistical or any other - by which we can
> find/calculate the value of Tangent of an angle without actually
> assigning values to perpendicular and base,
"perpendicular" is not a noun -- it does not have a "value".
> in which case triangle is
> not visualized?
Which triangle is 'not visulaized'?
> That is, if there is a very small gap in the curve
Curve? What curve?
> that shows the relation between 'A' (angle) and TanA, can we find the
> value of TanB, where B is an angle that lies in the gap?
Tan(a) ~= a is an approximation that will hold better the smaller a
is. Is that what you're asking for?
> Is Tangent of
> an angle 'unpredictable'?
There is nothing to "predict" here anywhere. If you have a value x,
then you can compute the value tan(x).
tan(x) = sin(x) / cos(x) ; where
sin(x) = lim( x - x^3/3! + ... ) ; and
cos(x) = lim(1 - x^2/2! + ... ) ; etc.
Douglas Eagleson
"V.Gopal" wrote:
The abstract angle A is a tangent's applied calculation in old
mathematics. The particular angle is counted so the relation of
the angle to the tangent equals a relation of the conic.
A- abstract angle
a-particular angle- a real significant digit one.
Where
a**2 + b**2 = 1 -the conic
and
1 = a/b - the tangent, relation of interest
A parametric where the solution to the tangent is an integer
where a and b are significant digit- integers. A transformation using
a factor of ten is always possible, creating a valid integer solution
to the conic relation. 1.000004 becomes 1000004 with no
effect on the solution.
The problem is the relation of the conic to the function of interest, a/b
is a Greek counting inference, also known as the relation of the
abstract tangent to the particular tangent. It is a lot of thinking, to
find
a particular unit integer of a or b where the other two relations are
satisfied.
One more transformation was added, the inference form where
the applied abstract tangent relation will cause the particular
tangent. And the relation of the abstract tangent is also the same
inference of counting applied to cause the abstract relation's
necessity, from your curious choice of a function of interest.
The relation a/b is independent of its usage for the definition
of a tangent. The ratio relation abstracted is the...., think hard.
Lots of work, thinking.
Abstract the necessity of a ratio for usage anywhere.
There are allot of angles out there, it seems. What else
in relation to the abstract number is a hard relation to define,
and this search for the suitability of the conic to most
number solutions in counting theory is defined by the
capacity to relate two or three variables as any number
such that the integer solution always appears, relative to the
relation of interest.
Again a sufficiency of the capacity to find a valid particular
counting solution is the only criteria for the usage of the conic.
So, the thinking of the abstract relation of the tangent was
only fruitful to test for the conics counting suitability.
Back to the definition of the counting inference.
The hard theory form is already stated and proved true
by unfailing success.
In effect either variable, a or b, must be transformed
as a unit rethinking. An iteration using a computer to
test an inference of counting, for your understanding of why
the abstract square root of two always equals about 1.4.
Significant digits always exist. This is another mathematics,
remember the test of its validity is the concept of the
abstract theory applied to the abstract number relationship.
A counting inference is always just another one.
And ineffect the value of the other two or three variables
are necessarily renormalized until the two relations agree!
And in the end the angle of the abstract angle applied using this
theory is always solved for a real measured applied usage
of a mathematics. No aesthetics, allowed. Works real
nice on a computer.
So your function of the tangent is solved always in relation to
the conic. And all real angles are provably integer.
Douglas Eagleson
Gaithersburg, MD USA
V.Gopal
> "V.Gopal" wrote:
> >
> > Is there any method - statistical or any other - by which we can
> > find/calculate the value of Tangent of an angle without actually
> > assigning values to perpendicular and base, in which case triangle is
> > not visualized? That is, if there is a very small gap in the curve
> > that shows the relation between 'A' (angle) and TanA, can we find the
> > value of TanB, where B is an angle that lies in the gap? Is Tangent of
> > an angle 'unpredictable'?
>
> TanA is a smooth and almost continuous function with countable
> singularities in known places. Linearly interpolate. If you want a
> better number, use Newtonian approximation.
many feel uneasy to discuss about waht is 'smooth'. I cannot
understand what do you mean by 'almost continuous'. If we do not know
how the 'countable singularities in known places' are 'connected' TanA
is not sm.. and continuous. Countable singularities open or endless.
Is the conventional A (angle) vs TanA curve correct? Can we consider
TanA as a variable? (Please see the message "what is a variable?")
"It is better to be vague and partly correct than to be precise and
completely wrong." Unfortunately a mathematician cannot take advantage
of this rule.
me...@cars3.uchicago.edu
>Uncle Al <Uncl...@hate.spam.net> wrote in message news:<3DAD8CFC...@hate.spam.net>...
>> "V.Gopal" wrote:
>> >
>> > Is there any method - statistical or any other - by which we can
>> > find/calculate the value of Tangent of an angle without actually
>> > assigning values to perpendicular and base, in which case triangle is
>> > not visualized? That is, if there is a very small gap in the curve
>> > that shows the relation between 'A' (angle) and TanA, can we find the
>> > value of TanB, where B is an angle that lies in the gap? Is Tangent of
>> > an angle 'unpredictable'?
>>
>> TanA is a smooth and almost continuous function with countable
>> singularities in known places. Linearly interpolate. If you want a
>> better number, use Newtonian approximation.
>"TanA is smooth!" ---- I found that 'smooth' is a disputable term and
>many feel uneasy to discuss about waht is 'smooth'. I cannot
>understand what do you mean by 'almost continuous'.
Continuous almost everywhere.
> If we do not know
>how the 'countable singularities in known places' are 'connected' TanA
>is not sm.. and continuous.
But we do know.
> Countable singularities open or endless.
That doesn't even mean anything.
>Is the conventional A (angle) vs TanA curve correct?
Sure, why not. Tan can be calculated to any required accuracy. Are
you familiar with the concept of Taylor series?
Mati Meron | "When you argue with a fool,
me...@cars.uchicago.edu | chances are he is doing just the same"
Edward Green
> In article <38af3945.02101...@posting.google.com>,
> V.Gopal <vgop...@rediffmail.com> wrote:
> >Is there any method - statistical or any other - by which we can
> >find/calculate the value of Tangent of an angle without actually
> >assigning values to perpendicular and base, in which case triangle is
> >not visualized?
>
>
Look up tables? :)
Hmm... maybe too memory intensive. But starting with Taylor series
seems too calculation intensive. Probably through some damn clever
and now standardized extrapolation.
Sam Wormley
>
> "TanA is smooth!" ---- I found that 'smooth' is a disputable term and
> many feel uneasy to discuss about waht is 'smooth'. I cannot
> understand what do you mean by 'almost continuous'. If we do not know
> how the 'countable singularities in known places' are 'connected' TanA
> is not sm.. and continuous. Countable singularities open or endless.
> Is the conventional A (angle) vs TanA curve correct? Can we consider
> TanA as a variable? (Please see the message "what is a variable?")
> "It is better to be vague and partly correct than to be precise and
> completely wrong." Unfortunately a mathematician cannot take advantage
> of this rule.
Don't tell me you don't know what a tangent function looks like!
Continuous [and smooth] with discontinuities at ą n pi/2 for
n = 1, 2, ...
Sam Wormley
"V.Gopal" wrote:
>
> "TanA is smooth!" ---- I found that 'smooth' is a disputable term and
> many feel uneasy to discuss about waht is 'smooth'. I cannot
> understand what do you mean by 'almost continuous'. If we do not know
> how the 'countable singularities in known places' are 'connected' TanA
> is not sm.. and continuous. Countable singularities open or endless.
> Is the conventional A (angle) vs TanA curve correct? Can we consider
> TanA as a variable? (Please see the message "what is a variable?")
> "It is better to be vague and partly correct than to be precise and
> completely wrong." Unfortunately a mathematician cannot take advantage
> of this rule.
Don't tell me you don't know what a tangent function looks like!
Continuous [and smooth] with discontinuities at ą n pi/2 for
n = 1, 3, 5, ...
puppe...@hotmail.com
> bwr...@u.washington.edu (Bryan Reed) wrote in message news:<aokasc$1ifc$1...@nntp6.u.washington.edu>...
>
> Look up tables? :)
>
> Hmm... maybe too memory intensive. But starting with Taylor series
> seems too calculation intensive. Probably through some damn clever
> and now standardized extrapolation.
If anybody is really interested, look up the key words
"predictor corrector" in a book on numerical methods.
Taylor expansion is only good for small angles or angles
close to special values like 45 degrees. For arbitrary
angles it tends to covnerge too slowly.
And you might be surprised how well look up tables can
be made to function when combined with some good
interpolation routines. Then, considering some good
trig identities, like tan = sin/cos, you don't need
the full set of trig look-ups.
Some machines even implement trig in hardware.
Socks
V.Gopal
perpendicular and base, p*b must be constant. This is a necessary
condition for DISPLAY of CONTINUITY. We can assume p=1, in which case
TanA=1/b, where 'b' is the continuos variable. 'b' decreases from
infinity to 1, when p=1. At p=b=1 there must be changein reference and
while b=1, p increases from 1 to infinity. This way the curve A
(angle) vs TanA looks like a hyperbola. If TanA*CotA=1 this symmetry
is logical. TanA does not have countable values. Tan0 to Tan45 and
Tan45 to Tan90 must be identical if TanA*CotA=1.
me...@cars3.uchicago.edu
>me...@cars3.uchicago.edu wrote in message news:<ORpr9.24$P4....@news.uchicago.edu>...
>> In article <38af3945.0210...@posting.google.com>, vgop...@rediffmail.com (V.Gopal) writes:
>>
>> >Is the conventional A (angle) vs TanA curve correct?
>>
>> Sure, why not. Tan can be calculated to any required accuracy. Are
>> you familiar with the concept of Taylor series?
>>
>If TanA=p/b is to be continuous then the product of the sizes of
>perpendicular and base, p*b must be constant.
Forget sizes, forget perpendicular, forget base. Tan A is a function
of the variable A. That's all. Where does A come from, this is
irrelevant.
And, you didn't answer my question. Are you familiar with the concept
of Taylor series.
... snip attempts at relevance ...
Douglas Eagleson
Douglas Eagleson wrote:
The astute math person will see the 1 on the two relations to be equated.
That is an abstract one!
So think of how an abstract one functions to cause an equation.
You will find a solution long before the 1 is found.
And it will always be found for a valid theory. And always be
found for a correct theory.
The relation is used here as a theory's relation of applied
mathematics. As an equation it is a counting function that solves.
The theory here would the theory of the angle's correct definition.
And the ratio of sides is the natural causes of all angles found in nature?
And here the concept of the subjective angle observed rather than the
causal angle is encountered.
Two sticks forming and angle symbol on the ground, is this a caused
natural angle? When can an angle be a symbol and a natural angle?
And the cause of the ratio in nature is found always in relation to
the necessity to measure an angle. A form of knowledge is found
in the definition to give the apperance as a relation of symbol
and applied symbol, as a means of causing angle in constructions.
Again the question is asked, does the causal angle exist?
And the search must be stated in relation to the theory
postulated.
So, use the applied theory of hot air rising. And a body
of hot air is seen to cause an angling of displaced air.
Does this angling appear as a symbol of a natural angle?
And the applied theory of inertia is soon seen to contain
an angle relation. There is a caused angle because of the
natural law of conservation of momentum. Scattering angles
consist of outcomes of theory applied.
So the conic was applied to allow the theory of many
such kenetic theory to be stated.
And then came quantum mechanics and superposition,
where the majic matrix of Hamilton allows a method of
matrix solution to state any form using the atom of the
differential.
Change of topic sorry. The concept of a solution by a
matrix method independent of theory is a foundation question.
What natural relation would allow such a majic method.
Hint, it is not anything aestethic, it is the relation
of the applied shift of any conserved natural quantity
to which an atom is found in theory.
Just like the concept of the angle the shift relation
is found in nature as a theory apparently true.
Except the conic will never solve for the atom.
Why is this, remember the capacity for solution is
defined as the applied form, and a cone is never
found as a conserved quantity. Not much of a reason
except the existence in nature is the fundamental proof
of any truth.
By craft of logic I can not make the the conic cause another conic!
Laurel Amberdine
> In article <38af3945.0210...@posting.google.com>, vgop...@rediffmail.com (V.Gopal) writes:
>
>> Countable singularities open or endless.
>
> That doesn't even mean anything.
Geez, Mati. You're *so* picky!
-Laurel
me...@cars3.uchicago.edu
would like to know what is the question.
Big Bird
> If TanA=p/b is to be continuous then the product of the sizes of
> perpendicular and base, p*b must be constant. This is a necessary
> condition for DISPLAY of CONTINUITY.
Dude, you need to lay off the drugs for a while.
A function f(x) is continuous in a point 'a' if and only if of every
series s_n that converges against a it is true that lim(f(s_n)) = f(a)
for s_n -> a