Talk:Exact trigonometric constants

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1 "No Solution"
2 Is it the same value?
3 pi/17
4 Factoring a fifth degree
5 Sin(1 degree)
6 pi / 7
7 Series calculations
8 Sine 20 degrees
9 nested radicals

[edit] "No Solution"

In the article, it says that cot 0˚ has "no solution". Should this be instead written "undefined"? Is there a difference? User:Error792

I don't think there's a difference. Tom Ruen 16:07, 23 June 2007 (UTC)

[edit] Is it the same value?

$4\sin 18^\circ = \sqrt{2(3 - \sqrt 5)} = \sqrt 5 - 1$

The last two expressions can be shown to be the same. Square the second one, and factor.

And my calculator, evaluates all three to the same value.

--MathMan64 02:02, 2 August 2005 (UTC)

I fixed this section to show how and when nested radicals can be simplified.--MathMan64 02:28, 8 September 2005 (UTC)

[edit] pi/17

It has been proved (I forgot who did it) that pi/17 takes an exact value and a regular 17-gon can be constructed with straight-edge and compasses only. Can anybody find more information about this and put it onto the table of constants? De ryck C. 08:43, 5 March 2006 (UTC)

See mathworld.wolfram.com/TrigonometryAnglesPi17.html

--MathMan64 20:22, 8 March 2006 (UTC)

[edit] Factoring a fifth degree

Yes I left out a factor of (y - 1). Thanks for finding it.

--MathMan64 20:17, 8 March 2006 (UTC)

[edit] Sin(1 degree)

I have heard that a formula for the sine of one third of an angle can be derived from the half angle and angle addtion indentities. It is something along the lines of 4sin(3x)^3-3sin(3x)-sin(x)=0. Does anyone know what the real formula is and should it be put in?Hiiiiiiiiiiiiiiiiiiiii 00:04, 2 May 2006 (UTC)

My assumption has been that this can't be solved in a radial form, but I've not looked into this. Tom Ruen 00:41, 2 May 2006 (UTC)

Well if there is a cubic equation with nontranscendental coefficients, it can be solved with radicals.Hiiiiiiiiiiiiiiiiiiiii 00:10, 4 May 2006 (UTC)

Sounds like a good challenge - to be proven or disproven. I started this article from math notes I kept from 20 years ago in my Trig class extra credit, never saw them printed up in any books. I'm happy if it can be shown to go further using closed cubic polynomial solutions. The nice thing about these are that they are pretty easy to test numerically on a calculator to confirm correctness. Tom Ruen 02:00, 4 May 2006 (UTC)

Well I found this on another wikipedia page. sin(3x)=3sin(x)-4(sin(x))^3. It appeared to be correct when I tested it with a calculator. Hiiiiiiiiiiiiiiiiiiiii 15:24, 6 May 2006 (UTC)

Okay, I see it at Trigonometric_identities#Triple-angle_formulae. Next step, solve that cubic equation for sin(x)=f(sin(3x)). Then test on something simple like sin(15)=f(sin(45)). If it works, you can work on sin(11), sin(10), sin(8), sin(7), sin(5), sin(4), sin(2), sin(1), etc, in any order you like. Tom Ruen 08:45, 22 May 2006 (UTC)

Actually, I'm working on this problem right now. The first two roots of the equation are the imaginary ones, so I'm going to start on the third. The speed of my work depends on my laziness, so I expect to have this done within the next week. CodeLabMaster 17:00, 6 June 2006 (UTC)

See my comment in the pi/7 section. 76.199.137.78 (talk) 03:43, 6 December 2007 (UTC)

[edit] pi / 7

I was doing some work with regular heptagons, which I now read are not ruler-and-compass constructible, and got to the fact that the cosine of pi/7 satisfies the cubic equation -

x^3 - x^2 - 2x + 1 = 0

where x = 2cos(pi/7), which is the ratio of the shorter diagonal to the sidelength in a regular heptagon.

Curiously enough, the ratio of the sidelength to the longer diagonal, which is sin(pi/7) / sin(3pi/7), is also a solution of this cubic - but not the same one.

I don't know if this means that the trig ratios for multiples of pi/7 should be included in this article or not. In a sense, this is an exact mathematical specification for the ratio as opposed to an approximation. On the other hand, from what I've read (today on wikipaedia) about solving cubics, I gather that actually computing these values requires reference back to trigonometric ratios anyway. - there is no expression in terms of surd-like expressions.

A useful reference is:

http://mathworld.wolfram.com/TrigonometryAnglesPi7.html

which gives sin & cos (and others) for pi/7, 2pi/7, 3pi/7 in terms of roots of polynomials.

Ben D R 08:19, 23 January 2007 (UTC)

Actually, a cubic (and even a quartic) can be solved in radicals, though it's hairy enough that usually nobody bothers to do so. So it would be possible to include explicit formulas for trig functions of arbitrary multiples of both pi/360 and pi/7. But they're still not constructible with straightedge and compass, because that only gives you a way to do square roots, not cube roots. The constructible numbers are a proper subset of the expressed-in-radical numbers, which are a proper subset of the algebraic numbers. 76.199.137.78 (talk) 03:43, 6 December 2007 (UTC)

[edit] Series calculations

How about showing how these constants can be calculated using the series definitions of the functions? That I would like to see. 163.1.62.82 (talk) 20:40, 13 January 2008 (UTC)