Possible Alternative to Trigonometry

I've been experimenting with this idea for a while now, because it's always confused me that there's no way to calculate stuff with trigonometry without either taking measurements in some way, or by using Pi to work out the value (which itself is calculated through measurements in some way).

I first worked on this a couple of years ago, but I gave up because I couldn't get past a certain problem I was having. But hopefully by posting here, maybe someone can figure it out better than me.

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Let's consider that this right-angled triangle's sides are 3, 4 and 5. This standard triangle has been proven to have the angles 90, 36.9 and 53.1.

A = 36.9
B = 53.1
C = 90

a = 3
b = 4
c = 5

We can think of the angles as being in 'parts'. Assuming we want to calculate angle A, we know that A must make up a portion of 180 - C (being the right angle). So A is part of 90.

The other part of the 90, is angle B. Now, to calculate one 'part', we do 90 / 7, where 7 is equal to the two affecting sides added together (a + b), (3 + 4). Because there are 7 parts in the 90, finding one part can be done by simply dividing by 7.

90 / 7 = 12.857

Now, to find out what A is, we just multiply by 3. This gives us 38.571. Of course, this doesn't equal 36.9. Try again with B, multiplying by 4. This gives us 51.428, which doesn't equal 53.1. You might notice though that the difference between the calculated value and the actual value is similar in both instances.

This is my problem:

When calculating an angle, the further away it is from 45 degrees, the bigger the difference. However, I can't seem to find a solid relationship between the two.

The formula that I have for this at the moment is:

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A = a / 180 - C \
      \  a + b  /

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When trying this with the standard 1, 1, root 2, triangle (with angles 45 45 and 90), calculating either of the 45 angles using the formula works out with exactly 45. So I know that it's orienting around 45 in some way.

If anyone can figure out a solution to this, it would be appreciated greatly.
Thanks,

ShockWave.

Did you hear what you said? That Trig is not an exact science. This is the first step to understanding that trig is not fit for formal presentations.

If you would like to really get into it, there is my work, The Delian Quest, and two more not yet posted one of which I am working with division of angles strickly through geometric methods. You will find DQ on my link. There is a lot there. I will say this, division of angles are an elliptic function. Easy to demonstrate.

In my work, there is no Trig, it is not a science. Nor, for the same reason, are there Cartesian Coordinate systems.

I have so many pokers in the fire, it I cannot keep up with what I am doing.

In my work, you will find a formula for any triangle, right triangles are simply a special case. The squares on any two sides of a triangle are equal to one half the square on the remaining side plus twice the square on the medial bisector.

Hm well, what you're kind of doing is, by assuming that you can split the angles up into parts proportional to the opposite side lengths, as in, a linear relationship... you're kind of just modelling the opposite edges as a circle.

It's just an estimation really. There's nothing particularly interesting about it.

The only good way to 'fix it' is to just do it the normal way with the standard trigonometric functions. With very few numbers will you be able to get a simple formula. Even in your example... the angles you gave were actually just rounded (not sure if you knew this), and are irrational numbers generated by an infinite series.

The trig functions do actually have some well known approximations, and they are based on estimating sides as circles.

To second order,

cosx = 1 - x^2 + E
sinx = x + E

I know that the 36.9 and other stuff were just rounded, but the example could easily have been the 1, 2, sqrt(3) triangle, with exact angles 30 and 60.

Aside from that, I did some thinking and searching, and I thought about when you said infinite series. Couldn't you just apply the MacLaurin Series to this, and add terms to get a more accurate value for sin/cos/etc x?

Yes, they are actually just the first three terms of the respective series.

You can see the sin one quite easily with a diagram; both of the approximations are most appropriate when x is small.

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Originally Posted by Xei

Hm well, what you're kind of doing is, by assuming that you can split the angles up into parts proportional to the opposite side lengths, as in, a linear relationship... you're kind of just modelling the opposite edges as a circle.

It's just an estimation really. There's nothing particularly interesting about it.

The only good way to 'fix it' is to just do it the normal way with the standard trigonometric functions. With very few numbers will you be able to get a simple formula. Even in your example... the angles you gave were actually just rounded (not sure if you knew this), and are irrational numbers generated by an infinite series.

The trig functions do actually have some well known approximations, and they are based on estimating sides as circles.

To second order,

cosx = 1 - x^2 + E
sinx = x + E

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You did not hear me at all. I said elliptical function, not linear. You cannot even imagine the figure. You learn geometry the same way you learn anything, learning to say what you see.

What I mean by this is that you have to spend thousands of hours constructing figures and saying them with algebra. I use two programs, Geometers Sketchpad and Mathcad.

There is a nice figure in my work that demonstrates that angle trisection, for examplle is directly related to a square root figure.

I did hear you but I wasn't talking to you, I was addressing the question posed by the original poster.

It's nice to see that you posted your little theorem; for the purposes of checking it, could I ask if you are referring to the bisector of the opposite sides, or the bisector of the angle?

You should know that all of this stuff although pretty is rather antiquated; it's easy to discover and to prove (or disprove) such theorems nowadays using more modern vector methods, and there's not really much use for them.

You have got to be joking.

No, sorry.

It seems my greatest flaw is overestimating other's conceptual abiity. Probably because I don't understand it.

If what you say is old news, why is non-Euclidean Geometry even taught? And why is Einstein still considered intelligent? And why is Trig anything more than something accountants use?

Well, thanks for the chat.

I... don't think accountants are in the habit of using geometry.

I really can't make head or tail of whatever argument you're making. I guess you misunderstood me, because I wasn't talking about non-Euclidean geometry or anything. The old news I was referring to was Euclidean Geometry. As I say, it's just rather antiquated. It was extremely important for the modern world historically because it reintroduced (in the Renaissance) the concept of rationality and infallible logical proof as authority rather than dogma, but in itself, it was more of a Greek preoccupation than anything. It is very beautiful and pure, and it is still studied for these reasons, but it is simply not necessary nowadays, as we have developed a hell of a lot more mathematics for dealing with Euclidean space, which simply makes the Euclidean methods redundant as it all falls out very simply of our more useful modern methods.

Of course Euclid's works were not wrong, they will remain true for time immemorial; it is just that we don't really have any practical interest in them nowadays. We learn a few essential, basic theorems in school, and then everything else follows.

I asked an extremely clear question in my previous post by the way, about a previous remark you made which was not sufficiently defined, and I'd appreciate it if you'd answer;

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Originally Posted by Xei

could I ask if you are referring to the bisector of the opposite sides, or the bisector of the angle?

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These are two different things and so you need to clarify.

Theoretical Disaster pt. 2?

Well, I suppose that A = A is old news, and somehow needs to be modernized. However, even Aristotle, as lame as he was, understood that logic, all logic is binary. You only have 2 choices from the very foundation of all logics, assertion and denial.
You only have 2 methods of constructing a set. Think they are related? And, since you only have 2 elements, you cannot predicate anything of them, even time. So, like it or not, they are neither old nor new. They just are.

And like it or not, There is only one flaw in Euclidean Geometry, which is easily rectified to make it elemental.

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Originally Posted by malac

Theoretical Disaster pt. 2?

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Ha, whilst that would be fun, I don't think I have the energy for it. If he keeps ignoring basic questions then I'll just let it lie again. Hopefully I've satisfied ShockWave.

Thanks Xei, the series stuff has helped a lot.

Also it was fun to watch this 'friendly debate', I almost went for popcorn :)

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Originally Posted by ShockWave

Thanks Xei, the series stuff has helped a lot.

Also it was fun to watch this 'friendly debate', I almost went for popcorn :)

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You should check out the 'Theoretical Disaster' thread, that one was epic. :)

Do you know much about Maclaurin series? They're really interesting... especially if you use them to consider exp(ix) in terms of cos(x) and sin(x).

You can also use the arctan series to get an expression for pi.

Can people stop the baloney?

True, the methods of calculating sin and cos and tan are not perfect and have some error, but you still need sin and cos and tan to do problems like these. The method of the threadstarter doesn't work for very big angles. It is only based off the small-angle approximation that sin x = tan x = x (in radians.)

Ya I already said that, this thread is kinda old and settled.