Triantánacht

Is brainse den mhatamaitic í an triantánacht a bhaineann leis na gaolta idir taobhanna is uillinneacha triantáin, bunaithe ar an bhfíoras go mbíonn dhá thriantán cosúil le chéile má bhíonn dronuillinn acu araon is uillinn amháin eile cosúil freisin. Sainmhínitear na feidhmeanna triantánúla mar choibhneasa na dtaobhanna i dtriantán dronuilleach. Is iad na feidhmeanna is coitinne, síneas (taobh urchomhaireach/taobhagán), comhshíneas (taobh cóngarach/taobhagán), agus tangant (taobh urchomhaireach/taobh cóngarach). Maidir le triantáin nach mbíonn dronuillinn iontu, is iad na torthaí is úsáidí ná an fhoirmle sínis a/sin A = b/sin B = c/sin C, agus an fhoirmle comhshínis a² = b² + c² – 2 b c cos A.^[1]

Stair

Rinne na Suiméaraigh,^[2] na Bablónaigh, agus na Núibigh ársa^[3] staidéar ar uillinneacha agus ar chóimheasa triantán, ach níor fhorbair siad riamh modh córasach chun triantáin a réiteach.

Chuaigh réalteolaithe na Gréige níos faide. Rinne Hioparcas na chéad táblaí corda timpeall 140 RC^[4] agus bhí táblaí Tolamaes san Almagest^[5] in úsáid sa réalteolaíocht ar feadh 1200 bliain ina dhiaidh sin.

Tháinig an síneas nua-aimseartha chun cinn sa Surya Siddhanta san India^[6] agus leathnaigh matamaiticeoirí Ioslamacha ar nós Abu al-Wafa^[7] agus Nasir al-Din al-Tusi an réimse, agus rinne al-Tusi disciplín ar leith den triantánacht, neamhspleách ar an réalteolaíocht.^[8]^[9]^[10]^[11]

Shroich an t-ábhar Iarthar na hEorpa trí aistriúcháin Laidine ar shaothair Ghréagacha agus Arabacha,^[12] rud as ar eascair De Triangulis le Regiomontanus^[13] agus sa bhliain 1595 chum Pitiscus an focal "triantánacht".^[14]

De bharr riachtanais na loingseoireachta agus na cartagrafaíochta, d'fhás an réimse ansin trí mhodh triantánaithe Frisius, an úsáid a bhain Euler as uimhreacha coimpléascacha^[15] agus sraitheanna triantánúla agus Taylor le Gregory, Maclaurin^[16] agus Taylor.^[17]

Feidhmeanna triantánúla

Léiríonn an tábla seo a leanas airíonna ghraif na sé fheidhm triantánúla:^[18]^[19]

Feidhm	Tréimhse	Fearann	Raon
síneas	$2\pi$	$(-\infty ,\infty )$	$[-1,1]$
comhshíneas	$2\pi$	$(-\infty ,\infty )$	$[-1,1]$
tangant	$\pi$	$x\neq \pi /2+n\pi$	$(-\infty ,\infty )$
seiceant	$2\pi$	$x\neq \pi /2+n\pi$	$(-\infty ,-1]\cup [1,\infty )$
comhsheiceant	$2\pi$	$x\neq n\pi$	$(-\infty ,-1]\cup [1,\infty )$
comhthangant	$\pi$	$x\neq n\pi$	$(-\infty ,\infty )$

Tagairtí

↑ Hussey, Matt (2011). "Triantánacht". Fréamh an Eolais. Coiscéim. p. 677.
↑ "Cambridge IGCSE Core Mathematics" (2018). Hachette UK. Sliocht den leathanach 275
↑ Otto Neugebauer (1975). "A history of ancient mathematical astronomy. 1". Springer-Verlag.
↑ Thurston (1996), pp. 235–236, "Appendix 1: Hipparchus's Table of Chords".
↑ Toomer, G. (1998), Ptolemy's Almagest, Nuachtáin Ollscoil Princeton, Bibcode:1998ptal.book.....T, ISBN 978-0-691-00260-6
↑ Boyer (1991), p. 215.
↑ Boyer 1991, p. 238.
↑ "Nasir al-Din al-Tusi". “One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.”
↑ "the cambridge history of science" (October 2013) 2: 62–83. Cambridge University Press. doi:10.1017/CHO9780511974007.004.
↑ "ṬUSI, NAṢIR-AL-DIN i. Biography". Encyclopaedia Iranica. Aisghafa 2018-08-05. His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.
↑ "trigonometry".
↑ Boyer (1991), pp. 237, 274.
↑ "Johann Müller Regiomontanus".
↑ Robert E. Krebs (2004). "Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance". Greenwood Publishing Group.
↑ Grattan-Guinness, Ivor (1997). "The Rainbow of Mathematics: A History of the Mathematical Sciences". W.W. Norton.
↑ Ewald, William Bragg (2005-04-21). "From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics" (as en). OUP Oxford.
↑ Dempski, Kelly (November 2002). "Focus on Curves and Surfaces" (as en). Premier Press.
↑ Mary P Attenborough (30 June 2003). "Mathematics for Electrical Engineering and Computing". Elsevier.
↑ "Calculus of a Single Variable" (10 November 2008). Cengage Learning.

[FreamhanEolais-1] Hussey, Matt (2011). "Triantánacht". Fréamh an Eolais. Coiscéim. p. 677.

[2] "Cambridge IGCSE Core Mathematics" (2018). Hachette UK. Sliocht den leathanach 275

[3] Otto Neugebauer (1975). "A history of ancient mathematical astronomy. 1". Springer-Verlag.

[FOOTNOTEThurston1996[httpsbooksgooglecombooksidrNpHjqxQQ9oCpgPA235_235–236]"Appendix_1:_Hipparchus's_Table_of_Chords"-4] Thurston (1996), pp. 235–236, "Appendix 1: Hipparchus's Table of Chords".

[toomer-5] Toomer, G. (1998), Ptolemy's Almagest, Nuachtáin Ollscoil Princeton, Bibcode:1998ptal.book.....T, ISBN 978-0-691-00260-6

[FOOTNOTEBoyer1991215-6] Boyer (1991), p. 215.

[FOOTNOTEBoyer1991238-7] Boyer 1991, p. 238.

[8] "Nasir al-Din al-Tusi". “One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.”

[9] "the cambridge history of science" (October 2013) 2: 62–83. Cambridge University Press. doi:10.1017/CHO9780511974007.004.

[10] "ṬUSI, NAṢIR-AL-DIN i. Biography". Encyclopaedia Iranica. Aisghafa 2018-08-05. His major contribution in mathematics (Nasr, 1996, pp. 208–214) is said to be in trigonometry, which for the first time was compiled by him as a new discipline in its own right. Spherical trigonometry also owes its development to his efforts, and this includes the concept of the six fundamental formulas for the solution of spherical right-angled triangles.

[Britannica-11] "trigonometry".

[FOOTNOTEBoyer1991237,_274-12] Boyer (1991), pp. 237, 274.

[13] "Johann Müller Regiomontanus".

[14] Robert E. Krebs (2004). "Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance". Greenwood Publishing Group.

[15] Grattan-Guinness, Ivor (1997). "The Rainbow of Mathematics: A History of the Mathematical Sciences". W.W. Norton.

[16] Ewald, William Bragg (2005-04-21). "From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics" (as en). OUP Oxford.

[17] Dempski, Kelly (November 2002). "Focus on Curves and Surfaces" (as en). Premier Press.

[Attenborough2003-18] Mary P Attenborough (30 June 2003). "Mathematics for Electrical Engineering and Computing". Elsevier.

[LarsonEdwards2008-19] "Calculus of a Single Variable" (10 November 2008). Cengage Learning.

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