Talk:Trigonometric tables

Untitled
Unfortunately, that's not a useful algorithm for generating sine tables, for a number of reasons. It will only work as the number of divisions tends towards infinity, with infinite-precision arithmetic.

I might add more algorithms, Ive asked a guy about incorporating his page with four algorthms into wikipedia. /sandos

(Has anyone ever actually used the Euler-integration method to compute trig tables?)

I would suggest dividing this article into a few sections:


 * Historical computation of trigonometric tables, before computers were widespread. Who did it?  What methods did they use?  How accurately did they compute them?


 * Recurrence algorithms used for FFTs, etcetera, summarizing the formulas that are most often used and their error characteristics.


 * Stap —Preceding unsigned comment added by 77.238.193.253 (talk) 12:00, 27 September 2008 (UTC)


 * Interpolation schemes that are used for employing tables to compute trig. functions of arbitrary arguments.

Probably, there should be a separate article on computing trigonometric functions, not necessarily tables per se, but how they are actually done in practice. (CORDIC algorithms, arithmetic-geometric mean techniques for arbitrary-precision arithmetic, etcetera.)


 * -- Steven G. Johnson

Moved
Moved from the article:
 * To come
 * Buneman's recurrence algorithm for accurate FFTs (Proc. IEEE 75, 1434 (1987)), or some similarly improved scheme (see Tasche, below).
 * Calculating accurate approximations for trigonometric functions (CORDIC schemes, etcetera)
 * Arbitrary-precision arithmetic methods (quadratically convergent schemes based on arithmetic-geometric mean, related to fast methods for computing pi)

&mdash; Timwi 16:24, 6 Mar 2004 (UTC)

Cool
You may please add some solved problems of trigonometry.{| class="wikitable" ! header 1 ! header 2 ! header 3
 * row 1, cell 1
 * row 1, cell 2
 * row 1, cell 3
 * row 2, cell 1
 * row 2, cell 2
 * row 2, cell 3
 * }
 * row 2, cell 3
 * }

The values of sine and cosine per every 675 seconds
Various combinations of wel known angles inserted into additional formule leads up to

the values of cosine and sine per every 3°.

$$\cos{3^\circ}=\sin{87^\circ}=\cos({18^\circ-15^\circ}) = \frac{1}{2}\sqrt{2 + \frac{\sqrt{3}+\sqrt{15}+\sqrt{10-\sqrt{20}}}{4}} $$

$$\sin{3^\circ}=\cos{87^\circ}=\cos({72^\circ+15^\circ}) = \frac{1}{2}\sqrt{2 - \frac{\sqrt{3}+\sqrt{15}+\sqrt{10-\sqrt{20}}}{4}} $$ The formulae will be shorter and easier to memorize introducing:

$$ 1.61803398874989484821 \mbox{ (Fibonacci) ≈ } \phi =2 \cos{36^\circ}=\frac{\sqrt{5}+1}{2} $$


 * $$\cos{3^\circ}=\sin{87^\circ}=\cos({18^\circ-15^\circ}) = \frac{1}{2}\sqrt{2 + \frac{\phi\sqrt{3}+\sqrt{3-\phi}}{2}} $$


 * $$\sin{3^\circ}=\cos{87^\circ}=\cos({72^\circ+15^\circ}) = \frac{1}{2}\sqrt{2 - \frac{\phi\sqrt{3}+\sqrt{3-\phi}}{2}} $$

Half-angle formulae applied 4 times give the values of cosine and sine for 675˝ = 11´15˝ = 3°/16:

$$\mbox{cos675˝} = \cos\frac{3^\circ}{2^4} = \frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2 + \sqrt{2+\sqrt{2+\frac{\phi\sqrt{3}+\sqrt{3-\phi}}{2}}}}}}   $$

$$\mbox{sin675˝} = \sin\frac{3^\circ}{2^4} = \frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2 + \sqrt{2+\sqrt{2+\frac{\phi\sqrt{3}+\sqrt{3-\phi}}{2}}}}}}  $$

The last is key factor of various Interpolation formulae. Sorry, Stepen - accidentally I removed your comment but accepting the suggestion. My idea was to add these few lines as an ilustration to the half-angle and angle-addition part of the article. 77.238.193.253 (talk) 21:30, 27 September 2008 (UTC)Stap


 * Re-added my comment from 1 October:
 * I'm not sure why you are working on this here. To the extent that it belongs in Wikipedia at all, it belongs in Exact trigonometric constants, not in this article.  But I'm not sure why we should include it at all; it doesn't seem to be particularly notable to construct more and more constants by applying the half-angle formula repeatedly, nor is this a practical algorithm any more on modern computers.  (Note, by the way, that your last formula seems to be off by a factor of 2.)  Also, if you are going to do extensive editing of Wikipedia, you should sign in and get a user name, to make it easier for other editors to communicate with you. And you should use LaTeX only for equations, not for general text. —Steven G. Johnson (talk) 23:27, 1 October 2008 (UTC)

Early History
Early trig tables such as that developed by Ptolemy used (unsuprisingly) Ptolemy's theorem although it is debatable whether "Ptolemy's Theorem" may in fact pre-date it's name! A point that should be noted is that both angle addition and subtraction formulae (for sine & cosine) as well as Pythagoras' theorem are - in effect - corollaries to or applications of Ptolemy's Theorem. In essence the method used was to first determine chords of basic angles (30,45,60), then chords of golden ratio related angles (18,36,54,72). With 36 and 30 determined, application of the difference method would give a 6 degree chord and thence by halves to 3 degrees and 1.5 degrees. The halving algorithm was a separate geometrical construct. All of which is described in great detail in Book I of Copernicus's DROC. Neither the Greeks, nor Copernicus ever used the terms 'sine', 'cosine' or the like - they worked entirely with chords of circles but the Copernican table of half chords is exactly a table of sine values.

Given which it is hard to understand why Ptolemy's Theorem seems to have more or less disappeared off the trigonometrical landscape. On the other hand Pythagoras theorem - a mere corollary (or "porism") thereof - appears in every last school Math text book.

Neil Parker (talk) 10:37, 3 August 2010 (UTC)

Casa
Casa 2A02:2F01:B207:A300:CC6B:A7F5:1C92:50C1 (talk) 05:16, 30 October 2022 (UTC)

Error in equation
The equation given in the last section ($$e^{i(\theta + \Delta)} = e^{i\theta} \times e^{i\Delta\theta}$$) contains an extraneous $$\theta$$ on the right-hand side, I believe. It should instead read $$e^{i(\theta + \Delta)} = e^{i\theta} \times e^{i\Delta}$$. Agreed? Kepsır (talk) 19:01, 8 May 2023 (UTC)