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'?
> 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.
-- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) "Quis custodiet ipsos custodes?" The Net!
In article <38af3945.0210160653.21dcc...@posting.google.com>,
V.Gopal <vgopa...@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?
vgopa...@rediffmail.com (V.Gopal) wrote in message <news:38af3945.0210160653.21dcc374@posting.google.com>... > 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.
"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'?
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.
> > 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'. 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.
In article <38af3945.0210161835.90b...@posting.google.com>, vgopa...@rediffmail.com (V.Gopal) writes: >Uncle Al <Uncle...@hate.spam.net> wrote in message <news:3DAD8CFC.8E70CE01@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"
bwr...@u.washington.edu (Bryan Reed) wrote in message <news:aokasc$1ifc$1@nntp6.u.washington.edu>... > In article <38af3945.0210160653.21dcc...@posting.google.com>, > V.Gopal <vgopa...@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...
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.
> "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 ± n pi/2 for n = 1, 2, ...
> "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 ± n pi/2 for n = 1, 3, 5, ...
> 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
> >> > 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"
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. 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.
In article <38af3945.0210171121.73122...@posting.google.com>, vgopa...@rediffmail.com (V.Gopal) writes:
>me...@cars3.uchicago.edu wrote in message <news:ORpr9.24$P4.1797@news.uchicago.edu>... >> In article <38af3945.0210161835.90b...@posting.google.com>, vgopa...@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 ...
Mati Meron | "When you argue with a fool, me...@cars.uchicago.edu | chances are he is doing just the same"
> > 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'?
> 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
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!
vgopa...@rediffmail.com (V.Gopal) wrote in message <news:38af3945.0210171121.73122584@posting.google.com>... > 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
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