Anyone know of a good web site or book that derives from the most basic first principles all of the relations between sin(theta) cos(theta) tan(theta) and how they are derived in relation to circles? I'm not just talking about how sin(theta) is opposite over adjacent, but WHY it is so. For instance, what calcualtion does the calculator do when you enter radians and then hit the sin or cosine button. How does it come up with the answer. Brian White
Very simply, the calculator starts with the same number (a complex number which has a magnitude and an angle and actually, one of a small set of complex numbers).
It knows the sine and cosine of that angle.
It then sees whether that angle is bigger or smaller than the angle you have input.
If bigger, it takes one number from a lookup table and multiplies the starting number by that, if smaller, it takes a different number and multiplies by that.
It also modifies the sine and cosine values it started with by other numbers in the same lookup table and derives the equivalent angle in radians.
The new angle will be nearer the value that was input than the first angle.
It then sees whether the new angle is bigger or smaller, uses the lookup table again and loops round again.
Eventually the angle will be near as dammit the angle that was input. It then has the sine and cosine and stops looping.
Or if you want a more precise explanation:
"Start with a unity-magnitude value of C = Ic + jQc. The exact value depends on the given phase. For angles greater than +90, start with C =
0 + j1 (that is, +90 degrees); for angles less than -90, start with C =
0 - j1 (that is, -90 degrees); for other angles, start with C = 1 + j0 (that is, zero degrees). Initialize an "accumulated rotation" variable to +90, -90, or 0 accordingly. (Of course, you also could do all this in terms of radians.) Do a series of iterations. If the desired phase minus the accumulated rotation is less than zero, add the next angle in the table; otherwise, subtract the next angle. Do this using each value in the table. The "cosine" output is in "Ic"; the "sine" output is in "Qc"."
For sin, cos or tan, etc, are functions that can be expressed in terms of real numbers (and are inherentl the ratio of real numbers), there is no need to add the complication of complex numbers as these are actually real number functions. Note that a computer or calculator actually handles complex numbers as pairs of real numbers which means that the use of complex numbers requires at least twice the computational effort involved with real numbers. Why bother. There are well established ways to do this a desired accuracy- these are based on summations of series and methods to truncate these. Any numerical analysis book discusses the theory under functional approximation with rational functions such as Chebyshev or Pade. The actual approach used may depend on the calculator but table look up does demand memory for the table and time to interpolate and even straight series summation to the desired accuracy works better in a limited range (which applies). For example: sin x (x in radians) =x-x^3/3!+x^5/5! -x^7/7!+..... and the convergence is quite fast . The factorial terms and the exponents are less of a problem than it appears as each term is the previous term multiplied by two numbers (7!=5!*6*7: x^7 =x^5*x^2 ) and a simple iterative approach is, while brute force, faster than table look-up. Table lookups such as you suggest are a throwback to the days of log tables where . Even if a table lookup is used, then interpolating polynomials can be used between data points so a table of, say, 10 value, is sufficient- this process was done in the log table using interpolation tables, but now we let the idiot savant do the number crunching.
Really understanding trigonometry involves all of these things. Calculus, complex exponentials, infinite series, differential equations, etc. That is what mathematical analysis is all about. Without that understanding, how would you get to Fourier series and integrals? Why should Bessel functions be related to trigonometric functions?
I use "Plane and Spherical Trigonometry", Bauer & Brook, D.C.Heath and Co., 1932.
On page 22, it says that sin, cos, tan, cot, sec and csc are _defined_ to be the ratio of the lengths of line segments. This will have to suffice for your "why" question.
Given enough computational horsepower, the values of the functions can be computed from their Taylor series expansions
scroll down). The formulas are difficult to compute, since factorial and x^n power can result in very large values. The good news is that you generally only need the first 3 or 4 terms to get a useful answer, but to get 10 digits of precision you need many terms.
Up until a few years ago, a hand calculator did not have a fast enough processor to perform these operations using the formula. Instead, values at certain points are computed ahead of time and stored in the calculator's program (say, for example, sin(1 degree), sin(2), sin(3), etc.).
When you enter your angle (for example, 2.7 degrees), the calculator interpolates between the values it knows (sin(2) and sin(3)). The calculator also takes advantage of how the functions repeat, since sin(1)=sin(361), and does that simpler calculation first. Changing radian, degrees or grads is also done first, and all computations are handled in radians.
------------- The original thread dealt with how trig functions were done with a calculator and that is what I answered.
"trigonometry" doesn't need an understanding of calculus or complex numbers. After all, the trig functions are just specific ratios of real numbers pairs describing a point on a unit circle with respect to the origin or center of the circle) and can be clearly explained well without the concepts of calculus, complex numbers, etc (Pythagorus is allowed as this is geometry). Certainly there is more but "trig" as it usually is considered is a basic building block in the understanding of complex numbers, Fourier series, etc. (specifically in terms of orthogonality) , not the other way around.
Sine is a definition. Because it is defined, it just is. BTW, sin is not opp/adj - that opp/adj value is tan, in a right triangle.
Mine (HP 48) does nothing. If you enter rads and then a value and then sin, it gives you the sin value for that angle.
I understand HP calculates the number using a series (the series is hard-programmed in the chip; it uses a series for the type of value you want: sin, cos, arcsin, hypersin, etc) - it only calculates to the level of accuracy on the display, BTW.
Are you saying that one needs to agree or disagree with another poster?
I was under the impression that facts were not matters of opinion; rather only the conclusions drawn from facts were matters of opinion.
1) Re: series. I speak only to most HP calculators - others may have stored tables rather than use a series. Did he make the same distinction, or is my post different?
2) Re" the "real" trig. I saw nothing in his posts as to the specific mathematic category of the trigonometric relationships, which goes to the core question and explains definitively the answer to the question of the original poster: where do I find a book on the real relationships of the trig functions? I could have just answered the original poster's question: "You don't" rather than giving the mathematical proof answer, i.e., "definition" means never having to explain why it is.
I know what Don Kelly posted. He even provided an example.
I never post any implications in questions. What I ask was the only answer I was seeking. I am not sure what you posted and wonder if you were replying to someone else. Meanwhile this question should be easy to answer: "So are you agreeing with what Don Kelly wrote, or saying something different?"
The only reason you could post:
is if you are assuming bias or facts not specifically posts. There is nothing more than a simple question.
"So are you agree>> So are you agreeing with what Don Kelly wrote, or saying
I do not remember, but I think the original poster wanted to "understand" trigonometric functions.
In any event, while the origin of trigonometry probably began from study of triangles, the trigonometric functions are so fundamental that they pervade all branches of mathematics. In principle, the study of vibration, including electrical vibration, could have been carried out without knowing about triangles. Infinite series or self replicating solutions of ordinary differential equations would lead to sines, cosines, real and complex exponentials, and a host of trigonometric identities.
The trig functions were defined originally, and still are, on the basis of ratios of sides of a right triangle. The usefullness mathematically in terms of vibration analysis, etc doesn't come from this relationship but from the variation of the sine and cosine values for a unit vector as it rotates through an angle with respect to a reference axis. This is the sort of thing that you are referring to i.e the solution to d^2x/dt^2 +ax =0 which happens to be expressible in terms of sine/cosine functions of t. Fair enough- however, the poor calculator, when x is given, and sin(x) is wanted, knows nothing of this and simply returns the geometric ratio for that particular x- which per se, is nothing to do with the solution of a DEQ, Euler etc. Note that vibration, Fourier series can also be expressed, more fundamentally, in terms of exponentials, without the use of sine or cosine.
By all this, I am not belittling the usefulness of trig functions in analysis- it's just that often the solutions to problems are most easily visualised in terms of trig functions (through the rotating vector, phasor or whatever).
OK. Now we are getting somewhere. How are you saying something different? Don Kelly suggested this is done with expansion series that tend to converge rather quickly. Is that how you are saying calculators do trig functions; or are you just guessing?
It usually is not that simple. While the usual Maclaurin series for trigonometric will converge relatively quickly for an angle of one radian, it will not work at all for 10000 let alone for a billion radians. The first step will be to take out multiple of 2*pi or more likely pi/4 in any implementation. There isn't enough memory available to used the infinite series for large arguments without invoking periodicity for trigonometric functions.
When tables were calculated manually, the usual approach was to make a lookup table for various steps and then use sum and difference formulas to get to a specific value. That may be done in a computer. There also are polynomial approximations available for that.
----------- All the range of conversion need be is 1.57.. radians so it makes sense to make use of periodicity so that convergence is fast -for sin, the terms in x,x^3, x^5..x^21 are enough and the related factorials 1,3! .. can be built on preceding terms. This is still a brute force approach but is better than table look-up and not as good as approximating polynomials which were well known in the applicable literature about 60 years ago (at least that is when I first ran across the topic).