What is the proper but basic way to find radians or angles ...

The Student said:
What is the proper but basic way to find radians or angles by just knowing two sides of a triangle
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Two sides do not define a triangle.

If you give me two lengths, I can make more triangles than you would care to look at. :cool:

PS: What are you thinking of, when you say "find radians or angles"?
 
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2 sides determine an angle

This is true only if you know something else about the triangle.
I suspect you are thinking of right triangles, because you mentioned Trig Functions.

The answer is that you can approximately measure angles in radians or degrees using a protractor.
If you are told the two sides of a right triangle (x,y), the ATan(y/x) gives the angle defined as accurately as you can calculate the ATan function.
 
Bob Brown MSEE said:
This is true only if you know something else about the triangle.
I suspect you are thinking of right triangles, because you mentioned Trig Functions.

The answer is that you can approximately measure angles in radians or degrees using a protractor.
If you are told the two sides of a right triangle (x,y), the ATan(y/x) gives the angle defined as accurately as you can calculate the ATan function.
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There must be a way to get the exact angle using calculus. In other words, there must be a known relationship between the arc length and the opposite side to the angle or radian size in question. In grade 12, we were taught how to find the arc length and angle by memorizing the unit circle diagram. I have to think that there is a really easy way to know the radians or angle of a right triangle that does not only have a height of 1/2, (2^1/2)/2 or (3^1/2)/2 with hypotenuses of one.
 
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The Student said:
There must be a way to get the exact angle using calculus. In other words, there must be a known relationship between the arc length and the opposite side to the angle or radian size in question. In grade 12, we were taught how to find the arc length and angle by memorizing the unit circle diagram. I have to think that there is a really easy way to know the radians or angle of a right triangle that does not only have a height of 1/2, (2^1/2)/2 or (3^1/2)/2 with hypotenuses of one.
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Why must the universe behave in accord with your presumptions about how it should? I suspect that the universe is equally oblivious to your intuitions as to mine. (Yes, it is a depressing thought.)

Furthermore, what do you mean by exact? Expressible as a rational number? Expressible in terms of trigonometric functions?
 
The Student said:
There must be a way
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Are you in that Egyptian river? ;)

There are ways to do things in which you see no trig functions, but the algebraic relationships are simply disguised forms of trigonometry (i.e., methods based on trig functions and then expressed algebraically by assigning symbols).

If you know sides a, b, and c, for example, there are "algebraic formulas" for the angles, but they are entirely based on sines, cosines, and tangents.

If you used no trigonometry to determine the angle between two given lengths, then you manually measured the angle. Gotta spare astrolabe laying around?

Cheers
 
JeffM said:
Why must the universe behave in accord with your presumptions about how it should? I suspect that the universe is equally oblivious to your intuitions as to mine. (Yes, it is a depressing thought.)
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If I expect the universe to work a certain way knowing my mathematical background, then I need psychological help - forget math.

Furthermore, what do you mean by exact? Expressible as a rational number? Expressible in terms of trigonometric functions?
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Well, then what does the calculator function cotan do to the ratio of the triangle's sides? whatever that is is probably what I am trying to figure out.
 
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The Student said:
Well, then what does the calculator function cotan do
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Most scientific calculators are programmed to start adding terms of an infinite sequence (these terms are algebraic expressions derived from trigonometry), the sum of which increasingly approaches the trig function at hand. It continues adding terms until its result reaches the precision threshold set by the manufacturer or user.

Same thing happens, with built-in functions to determine angles from sides.


EG:

When you know sides a, b, and c, of some triangle, and you want angle A, then you'd enter the sides, and the calculator's programming would do something like this:

x = (b^2 + c^2 - a^2)/(2bc)


A = Pi/2 - x - 1/2*x^3/3 - 1/2*(3/4)*x^5/5 - 1/2*(3/4)*(5/6)*x^7/7 - and so on … until 16-digits' precision (or whatever) is attained.


The formulas are based on trigonometry. :cool:
 
The Student said:
Well, then what does the calculator function cotan do to the ratio of the triangle's sides? whatever that is is probably what I am trying to figure out.
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It uses one of several methods of approximation. As mmm said, the approximation is as good the manufacturer decides to make it, but it seldom is exact.
 
mmm4444bot said:
Most scientific calculators are programmed to start adding terms of an infinite sequence (these terms are algebraic expressions derived from trigonometry), the sum of which increasingly approaches the trig function at hand. It continues adding terms until its result reaches the precision threshold set by the manufacturer or user.

Same thing happens, with built-in functions to determine angles from sides.


EG:

When you know sides a, b, and c, of some triangle, and you want angle A, then you'd enter the sides, and the calculator's programming would do something like this:

x = (b^2 + c^2 - a^2)/(2bc)


A = Pi/2 - x - 1/2*x^3/3 - 1/2*(3/4)*x^5/5 - 1/2*(3/4)*(5/6)*x^7/7 - and so on … until 16-digits' precision (or whatever) is attained.


The formulas are based on trigonometry. :cool:
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Ok, so when people are given the lengths of all 3 sides of a triangle, is it possible to find an exact value for its radians?
 
The Student said:
when people are given the lengths of all 3 sides of a triangle, is it possible to find an exact value for its [angles]?
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Not an exact value; an approximation to as many decimal places of precision that they desire. (Your original post does not mention "exact values" for angle measurements; is that what you're thinking?)
 
mmm4444bot said:
Not an exact value; an approximation to as many decimal places of precision that they desire. (Your original post does not mention "exact values" for angle measurements; is that what you're thinking?)
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Yeah, I should have asked for the exact value in the OP.
 
What bothers me about this question is that the OP is using "radian" as synonym of "angle".

It is like using "centimeter" as synonym of "length"!!

Or using "Newton" as synonym of "force"!!
 
Subhotosh Khan said:
What bothers me about this question is that the OP is using "radian" as synonym of "angle".

It is like using "centimeter" as synonym of "length"!!

Or using "Newton" as synonym of "force"!!
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Aren't they interchangeable?

I can say 180 degrees or 3.14 radians. But I can't say 5 force or 5 Newtons, right?
 
It was a typo, then. You titled this thread "find radians or angles" when you meant "find radians or degrees".

Also, irrespective of how a calculator works, when find an arctangent of a ratio of two sides, you are assuming a right angle between the two sides so you are assuming one angle.
 
There is a good reason that you see no simple explaination and definition.
Angles are theoretically unsound (Click here) until a basis for transendental numbers and circular (Trig) functions are established. It is possible to develope a complete triginomatry without using angles or sin cos, tan, etc.

Having said that, let's start from scratch.
You are a Greek student of Archimedes. We will call you Student. It is your job to invent a way to communicate the size of an angle. Well, you could invent a standard ruler that is curved -- that is very practical, but not fundamental. You might draw on your drawing skills. You know that if you make a perfect copy (on your drawing pad) of a large painting -- then every thing will be smaller. How much? Maby, you will choose 1/10. EVERYTHING IS DIFFERENT. The lengths are all 1/10 the measure on the original painting.

WAIT A MINUTE. Something is the same! If I take any 2 lines in the painting and calculate their ratio, I get the EXACT same Ratio in my drawing. ALL division problems that I can think of on the painting will yield the same result done on my drawing.

RULE: WOW we have a rule... only division (proportions) are the same between the painting and the drawing! Archimedes will be so proud of me when I show him this new rule.

OOPs: Archimedes liked the rule except for one thing ... it is wrong. Division isn't the only similarity. ALL ANGLES MATCH exactly too!!!!!
Student had to go to his bath and think. ("you can do your best thinking in the baths" -- Eureka)

ANSWER: I'll define angles by dividing the opposite side (perpendicular) by the distance to a perpendicular. (you said I could invent a way to measure angles).

Archemedies: "Darn"
Your rule IS correct. Angles are but a Corollary.

===========================

So angles are best thought of as ratios. However, the arc-tangent function was invented so that (angle a) + (angle b) = angle (a+b)
 
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Untitled.gif
θ is the angle (in radians)
r is the radius
L is the length of the arc cut by the lines (rays) that make the angle.

Then:
θ = L/r

This appears to be a simple definition, but the length of L is a very complicated concept!
 
More info is needed than 2 sides unless it's a right triangle. Non right triangles contain right triangles from which you can get Sine, Cosine, etc. if you know 1 more side or angle. Cosine is just the quotient of the perpendicular line adjacent to the angle & the hypotenuse.
 
trinzed said:
More info is needed than 2 sides unless it's a right triangle. Non right triangles contain right triangles from which you can get Sine, Cosine, etc. if you know 1 more side or angle. Cosine is just the quotient of the perpendicular line adjacent to the angle & the hypotenuse.
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Hi trinzed,

You are correct. But...
L is an arc, not the side of a triangle.

This drawing illustrates how a central angle can be measured by the length of the subtended arc. In this case, the units are called "radians" because the angle is the number of radii that are represented in the arc.
 
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