Talk:Chord (geometry)

Merge to secant line?

[For context, special:diff/27630665 proposed a merge with secant line]

I don't think this should be merged into secant line; this concept and in particular this word are far too prevalent and often used in different contexts. The proposition about "power of a point" does not fit into the secant line article. Michael Hardy 19:40, 7 November 2005 (UTC)

I second this disagree. A chord is always related to a circle, whereas a secant line is not. Also, "chord" in this context was an early trigonometric function, first tabulated by Hipparchus. Once there's a history section on this page, I think the differences should become clear. --Dantheox 20:32, 23 December 2005 (UTC)

I'm going to go ahead an remove the merge banner, as nobody has supported this merge on either page, and the additions I've made to this page make it a pretty transparently bad idea. --Dantheox 06:02, 24 December 2005 (UTC)

Another point, should the trigonometric stuff be in a separate article, possibly Chord (trigonometry)? The two are obviously related, but I'm not sure if the relationship is strong enough to put the two concepts (geometry and trig) on the same page. --Dantheox 06:09, 24 December 2005 (UTC)

Formula

I have a different issue than the underlying paragraphs. is there a formula that can tell the length of a chord? for you given information you know that the chord's endpoints are endpoints of an arc, you know the length and measure of the arc (the measure of the arc is not necassarily 90° and not necessarily 45°) as well of the radius of the circle. Is there a formula to tell me the length of this chord? If so, what is it? —The preceding unsigned comment was added by 24.187.129.93 (talk) 19:09, 26 April 2007 (UTC).

If the angle is θ, and the radius is r, then the chord if r*crdθ. (crd=2sin). The area cordoned off by the chord is (take the pie piece minus the triangle):

$\pi r^{2}{\frac {360}{\theta }}-\sin \theta {\frac {\sqrt {\theta ^{2}-r^{2}\sin \theta ^{2}}}{2}}$ 24.208.253.57 00:22, 11 October 2007 (UTC)

The preceding formula in answer to the above question how to determine the value of crd θ seems to beg the question since crd θ is used to define the function. Does a method exist ( absent the obvious sin(X) relation since it didn't exist at the time this function was used) to arrive at the value of crd θ to determine the length of the Chord?

In other words find crd θ and don't use crd() without explaining how to start evaluating it at certain points and don't use sin(). Provide a definition that renders crd() useful without dependence on sin() or other trig functions. How did the ancient mathematicians actually construct a chord table and render a value for this function?

For example take a central angle θ with r having a known length of 1. How can I determine r * crd(θ), given only θ and r? —Preceding unsigned comment added by 99.152.64.170 (talk) 02:39, 5 August 2009 (UTC) --99.152.64.170 (talk) 02:42, 5 August 2009 (UTC)--Steve99.152.64.170 (talk) 02:42, 5 August 2009 (UTC)--99.152.64.170 (talk) 02:42, 5 August 2009 (UTC)

Trig One

The chord version of the trig one rule is wrong, when i put it into calculators it gives 4 not 1 —Preceding unsigned comment added by 217.208.246.94 (talk) 13:17, 28 March 2008 (UTC)

Crd x = 2*Sin(x/2)

that means Sin x = 1/2 * Crd 2x

and since Sin(90°-x)=Cos x

that means Cos x = 1/2 * Crd(2*(90°-x)) = 1/2 * Crd(180°-x)

trigometric one says

Sin² x + Cos² x = 1

if we replaces with the chords we get

(1/2 * Crd 2x)² + (1/2 * Crd(180°-2x))² = 1

simplyfy:

Crd²(2x) / 4 + Crd²(180°-2x) / 4 = 1

multiplying with 4 we get

Crd²(2x) + Crd²(180°-2x) = 4

since X doesnt matter we get

Crd²(x) + Crd²(180°-x) = 4

which also can simply be proven by putting x = 0 or 180°, one of them is going to equal 2 then, and 2 squared = 4 hence it can not again be 1

it is NOT 1 proven

—Preceding unsigned comment added by 217.208.246.94 (talk) 13:17, 4 April 2008 (UTC) 

Yeah, I'm almost certain that the trigonometric formula for crd(θ) in terms of sin(θ) under the "Chords in trigonometry" section of the article is wrong. It claims that crd(θ)=2*sin(θ/2). I don't think this can possibly be right because sin(θ/2) is negative for some values of θ/2, and, therefore, it is negative for some values of θ, and, since a number that's twice a negative number is also negative, 2*sin(θ/2), which the article claims equals crd(θ), must also be negative for some values of θ. For example, 2*sin(1/2), with 1 as θ, is negative regardless of whether degrees or radians are used. However crd(θ) can't be negative for any real value of θ because, geometrically, crd(θ) is a distance, and, algebraically, it is the output of a square root, a fact that is mentioned in the very same equation! However, it does seem that crd(θ)=|2*sin(θ/2)|. Probably, someone discovered that (crd(θ))²=(2*sin(θ/2))² and proceeded to take the square root of both sides of the equation, entirely forgetting that the square root of a real number's square is the absolute value of that number! I normally refrain from editing Wikipedia articles, instead confining myself to talk pages, but seeing as this is a clear factual error, I'm going to remove the 2*sin(θ/2) from the end of that equation, leaving it at crd(θ)=√((1-cos(θ))²+(sin(θ))²)=√(2-2*cos(θ)). I don't see how this can possibly introduce erroneous information that wasn't already there into the article, and it has a significant chance of improving it. If you can actually prove that crd(θ)=|2*sin(θ/2)|, please put that in, and if I'm just missing something and am wrong, please revert my change.

69.112.209.47 (talk) 21:53, 11 June 2018 (UTC)

It seems that the statement about the arcchord or acrd function at the end of the article is probably wrong as well. I won't delete it, but I'll add a superscript tag to it. 69.112.209.47 (talk) 22:07, 11 June 2018 (UTC)

What you missed is that the angle is the angle subtended by the chord and so must lie in the interval 0 < θ ≤ 180°, so sin θ is non-negative. This angle is always taken in the positive sense. Perhaps this needs to be made more explicit in the article. --Bill Cherowitzo (talk) 03:57, 12 June 2018 (UTC)

Is a circle's diameter a chord...

or is a chord only a non-diameter connector? The definition should make this explicit. —Preceding unsigned comment added by 24.73.165.122 (talk) 23:44, 26 May 2009 (UTC) hello:)

No such caveat appears, and there's nothing in the article that suggests it. You can assume that either the diameter line is a chord or that the article needs to be edited. Google it (result 1) or Wikipedia it (line 2 as of this edit). Articles do not need to contain caveats about the absence of a caveat. ᛭ LokiClock (talk) 02:12, 24 March 2010 (UTC)

The second property clearly states that "Chords can not go through the center of a circle". This obviously conflicts with the definition of a chord at the start of the article "a chord is a line segment joining two points on any curve" and also with the section on chords in the article on circles "The diameter is the longest chord of the circle". It should say that the longest chord of a circle is a diameter or a chord that goes through the centre of a circle is a diameter. Ted (talk) 14:51, 12 January 2013 (UTC)

Is there any Calculus involving chords?

I was wondering cause I have a Calculus class next semester and I suck at chords? Thanks.68.156.142.92 (talk) 16:50, 3 February 2011 (UTC)

TODO

Remember that Hipparchus was the inventor of this method — Preceding unsigned comment added by Madsmtm (talk • contribs) 22:44, 1 June 2016 (UTC)

Defining Variables

Would it be helpful to define the variables that are used in the table? — Preceding unsigned comment added by Adel314 (talk • contribs) 13:01, 13 October 2019 (UTC)

In the table 'a' is not defined.

You made a table and introduced the parameter "a" and never defined what "a" was. 2600:1700:7890:5A40:0:0:0:49 (talk) 14:26, 17 January 2023 (UTC)

trigonometric chord function defined as double the sine of half the angle seems anachronistic

Through the entire period of chord-based trigonometry, chords were calculated based on some non-unit sinus totus (maximum value of the sine, i.e. the circle radius), and were thought of as representing line segments more than numbers. Writing it in modern notation and based on unit-circle trigonometry seems like an anachronism and somewhat misleading to readers. –jacobolus (t) 21:22, 13 November 2023 (UTC)

I tend to agree (and one of the references is to a FORTRAN code?!). I'm not sure we need to say anything other than that in a unit circle, the chord length is twice the sine of half the angle. The notation in this section looks like someone pushing their pet enthusiasm for doing trigonometry with chords instead of sines. Maybe that wasn't the original motivation, but it's how it reads now. XOR'easter (talk) 22:52, 13 November 2023 (UTC)

"Chord (math)" listed at Redirects for discussion

The redirect Chord (math) has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 March 26 § Chord (math) until a consensus is reached. 1234qwer1234qwer4 13:49, 26 March 2024 (UTC)