Trigonometria

Ny trigonometria dia sampan' ny matematika mandinika ny fifandraisan' ny zoro sy ny halavan' ny lafy amin' ny telozoro. Avy amin' ny teny grika tranainy hoe τρίγωνον, midika hoe « telozoro », sy μέτρον, midika hoe « refy », ny anarany.^[1] Ny lefa trigonometrika, indrindra indrindra, dia mampifandray ny zoro amin' ny telozoro misy zoro mahitsy sy ny tahan' ny halavan' ny lafiny.

Nipoitra tao amin' ny tontolo helenistika tamin' ny taonjato faha-3 talohan' i Kristy io sampam-pahaizana io, avy amin' ny fampiharana ny jeômetria tamin' ny fandinihana ny zavatra eny amin' ny lanitra.^[2]^[3] Nifantoka tamin' ny kajy momba ny ampahan-kitsy mampifandray teboka roa eo amin' ny faribolana ny Grika, fa ny mpahay matematika tany India kosa no namorona ny tabilao tranainy indrindra fantatra mirakitra ny sandan' ny taha trigonometrika, izay antsoina koa hoe lefa trigonometrika. Anisan' izany ny sine, izay tahan' ny halavan' ny lafy mifanatrika amin' ny zoro iray amin' ny halavan' ny lafy mifanatrika amin' ny zoro mahitsy, ao amin' ny telozoro misy zoro mahitsy.^[4]

Nandritra ny tantara dia nampiharina tamin' ny sehatra toy ny fandrefesana ny endrika sy ny refin' ny Tany, ny fandrefesana tany, ny fandalinana ny fihetsiky ny zavatra eny amin' ny lanitra araka ny lalàn' ny mekanika, ary ny famaritana toerana sy lalana amin' ny fivezivezena ny trigonometria.^[5]

Mampiavaka ny trigonometria ny fananany fitoviana matematika maro izay marina amin' ny sanda rehetra azo raisina. Ireo fitoviana trigonometrika ireo^[6] dia ampiasaina matetika hanoratana indray ny fanehoana matematika trigonometrika, mba hanatsorana azy, hahitana endrika mahasoa kokoa amin' ilay fanehoana, na hamahana fampimirana.^[7]

Taha trigonometrika

Ny taha trigonometrika dia ny tahan' ny halavan' ny lafy amin' ny telozoro misy zoro mahitsy. Miankina amin' ny zoro iray latsaky ny zoro mahitsy ao amin' ilay telozoro ihany ireo taha ireo, satria mitovy endrika ny telozoro roa misy zoro mahitsy raha mitovy ny zoro iray latsaky ny zoro mahitsy ao aminy.^[8]

Noho izany dia mamaritra lefa miankina amin' io zoro io ireo taha ireo; lefa trigonometrika no iantsoana azy. Voafaritra eto ambany ireo lefa ireo arakaraka ny zoro A efa fantatra, ka ny a, b ary h dia manondro ny halavan' ireo lafy eo amin' ny sary.

Amin' ireto famaritana manaraka ireto, ny lafy lava indrindra dia ilay lafy mifanatrika amin' ny zoro 90° ao amin' ny telozoro misy zoro mahitsy. Io no lava indrindra amin' ny lafin' ilay telozoro ary iray amin' ireo lafy roa mifanila amin' ny zoro A. Ny lafy mifanila kosa dia ilay lafy iray hafa mifanila amin' ny zoro A. Ny lafy mifanatrika dia ilay lafy mifanatrika amin' ny zoro A. Indraindray dia antsoina hoe lafy mijadona (perpendicular) ny lafy mifanatrika, ary lafy iorenana (base) ny lafy mifanila. Hazavaina etsy ambany ao amin' ny fomba fitadidiana ireo anarana ireo.

Ny sine, soratana hoe $\sin$ , dia ny tahan' ny halavan' ny lafy mifanatrika amin' ny zoro amin' ny halavan' ny lafy lava indrindra.

\sin A = \frac{lafymifanatrika}{lafylavaindrindra} = \frac{a}{h} .

Ny cosine, soratana hoe $\cos$ , dia ny tahan' ny halavan' ny lafy mifanila amin' ny zoro amin' ny halavan' ny lafy lava indrindra. Ilay lafy mampifandray ny zoro dinihina amin' ny zoro mahitsy no lafy mifanila eto.

\cos A = \frac{lafymifanila}{lafylavaindrindra} = \frac{b}{h} .

Ny tangent, soratana hoe $\tan$ , dia ny tahan' ny halavan' ny lafy mifanatrika amin' ny halavan' ny lafy mifanila.

\tan A = \frac{lafymifanatrika}{lafymifanila} = \frac{a}{b} = \frac{a / h}{b / h} = \frac{\sin A}{\cos A} .

Ny sanda mifamadika amin' ireo taha ireo, izany hoe 1 zaraina amin' ny taha tsirairay, dia antsoina hoe cosecant ( $\csc$ ), secant ( $\sec$ ) ary cotangent ( $\cot$ ), araka izany filaharana izany:

\csc A = \frac{1}{\sin A} = \frac{lafylavaindrindra}{lafymifanatrika} = \frac{h}{a},

\sec A = \frac{1}{\cos A} = \frac{lafylavaindrindra}{lafymifanila} = \frac{h}{b},

\cot A = \frac{1}{\tan A} = \frac{lafymifanila}{lafymifanatrika} = \frac{\cos A}{\sin A} = \frac{b}{a} .

Ny cosine, cotangent ary cosecant dia nomena ireo anarana ireo satria izy ireo no sine, tangent ary secant, araka izany filaharana izany, an' ilay zoro izay mahafeno zoro mahitsy rehefa ampiana ny zoro dinihina. Ny hoe co- dia fanafohezana ilazana io fifamenoan' ny zoro io.^[9]

Amin' ny fampiasana ireo lefa ireo sy ny fitsipika momba ny sine ary ny fitsipika momba ny cosine, dia azo valiana saika ny fanontaniana rehetra momba ny telozoro na inona na inona endriny.^[10] Azo ampiasaina ireo fitsipika ireo mba hikajiana ny zoro sy ny lafy sisa amin' ny telozoro iray raha fantatra ny lafy roa sy ny zoro eo anelanelany, na ny zoro roa sy ny lafy iray, na ny lafy telo.

Fomba fitadidiana

Fampiasana mahazatra ny fomba fitadidiana ny fampahatsiahivana ny zava-misy sy ny fifandraisana ao amin' ny trigonometria. Azo tsaroana, ohatra, ny taha sine, cosine ary tangent amin' ny telozoro misy zoro mahitsy, amin' ny alalan' ny filaharan-tarehintsoratra maneho azy ireo sy ny lafy mifandraika aminy. Anisan' izany ny SOH-CAH-TOA (fanafohezana anglisy mampahatsiahy ny lafy zaraina amin' ny lafy hafa amin' ireo taha telo):^[11]

Sine = Opposite ÷ Hypotenuse — ny sine dia ny halavan' ny lafy mifanatrika zaraina amin' ny halavan' ny lafy lava indrindra.

Cosine = Adjacent ÷ Hypotenuse — ny cosine dia ny halavan' ny lafy mifanila zaraina amin' ny halavan' ny lafy lava indrindra.

Tangent = Opposite ÷ Adjacent — ny tangent dia ny halavan' ny lafy mifanatrika zaraina amin' ny halavan' ny lafy mifanila.

Fomba iray ahatsiarovana ireo tarehintsoratra ny fanononana azy ireo ho toy ny teny iray: /ˌsoʊkəˈtoʊə/, na SOH-kə-TOH-ə, izay mitovitovy amin' ny fanononana ny hoe Krakatoa.^[12] Fomba iray hafa ny fampiasana ireo tarehintsoratra ireo ho fiandohan' ny teny ao amin' ny fehezanteny iray, toy ny hoe “Some Old Hippie Caught Another Hippie Trippin' On Acid” (izany hoe, amin' ny teny malagasy: nisy olona antitra anisan' ny hetsika ara-kolontsaina hippie nahatratra olona iray hafa tao amin' io hetsika io, izay teo ambany fiantraikan' ny zava-mahadomelina antsoina hoe acid).^[13]

Faribolana manana elanelana 1 amin' ny foibe sy sanda trigonometrika mahazatra

Azo aseho amin' ny alalan' ny faribolana manana elanelana 1 amin' ny foibe koa ireo taha trigonometrika. Io dia faribolana ao amin' ny lemaka, izay eo amin' ny fiandohan' ny rafitra fangitra ny foibeny, ary mirefy 1 ny tapaka mampifandray ny foibe amin' ny teboka tsirairay eo amin' ny faribolana.^[14] Amin' io fanehoana io, ny lafy faran' ny zoro A, izay apetraka amin' ny toerana fanovozana mahazatra ny zoro, dia mifanojo amin' ilay faribolana eo amin' ny teboka (x,y), ka $x = \cos A$ ary $y = \sin A$ .^[14] Io fanehoana io dia ahafahana mikajy ireo sanda trigonometrika fampiasa matetika, toy ireo ao amin' ity tabilao manaraka ity:^[15]

Sanda ampidirina (radiana)	0	$π / 6$	$π / 4$	$π / 3$	$π / 2$	$2 π / 3$	$3 π / 4$	$5 π / 6$	$π$
Sanda ampidirina (degre fandrefesana zoro)	0°	30°	45°	60°	90°	120°	135°	150°	180°
sine	$0$	$1 / 2$	$\sqrt{2} / 2$	$\sqrt{3} / 2$	$1$	$\sqrt{3} / 2$	$\sqrt{2} / 2$	$1 / 2$	$0$
cosine	$1$	$\sqrt{3} / 2$	$\sqrt{2} / 2$	$1 / 2$	$0$	$- 1 / 2$	$- \sqrt{2} / 2$	$- \sqrt{3} / 2$	$- 1$
tangent	$0$	$\sqrt{3} / 3$	$1$	$\sqrt{3}$	Tsy voafaritra	$- \sqrt{3}$	$- 1$	$- \sqrt{3} / 3$	$0$
secant	$1$	$2 \sqrt{3} / 3$	$\sqrt{2}$	$2$	Tsy voafaritra	$- 2$	$- \sqrt{2}$	$- 2 \sqrt{3} / 3$	$- 1$
cosecant	Tsy voafaritra	$2$	$\sqrt{2}$	$2 \sqrt{3} / 3$	$1$	$2 \sqrt{3} / 3$	$\sqrt{2}$	$2$	Tsy voafaritra
cotangent	Tsy voafaritra	$\sqrt{3}$	$1$	$\sqrt{3} / 3$	$0$	$- \sqrt{3} / 3$	$- 1$	$- \sqrt{3}$	Tsy voafaritra

Lefa trigonometrika amin' ny miova tena na haro

Amin' ny fampiasana ny faribolana manana halavan-tapaka iray avy amin' ny foibe, dia azo itarina amin' ny sanda rehetra ampidirina amin' ny lefa, na miabo (lehibe noho ny aotra) na miiba (kely noho ny aotra), ny famaritana ireo taha trigonometrika.^[16] Momba izany no resahin' ny lefa trigonometrika.

Kisary maneho ny lefa trigonometrika

Ity tabilao manaraka ity dia mamintina ny toetran' ny kisary maneho ireo lefa trigonometrika fototra enina.^[17]^[18] Ny fitsingerenana dia ny fitomboana miabo kely indrindra amin' ny sanda ampidirina izay mamerina ny sandan' ny lefa.

Lefa	Fitsingerenana	Vondron' ny sanda azo ampidirina	Vondron' ny sanda azo avoaka
sine	$2 π$	$(- \infty, \infty)$	$[- 1, 1]$
cosine	$2 π$	$(- \infty, \infty)$	$[- 1, 1]$
tangent	$π$	$x \neq π / 2 + n π$	$(- \infty, \infty)$
secant	$2 π$	$x \neq π / 2 + n π$	$(- \infty, - 1] \cup [1, \infty)$
cosecant	$2 π$	$x \neq n π$	$(- \infty, - 1] \cup [1, \infty)$
cotangent	$π$	$x \neq n π$	$(- \infty, \infty)$

Lefa mifamadika amin' ny lefa trigonometrika

Satria miverimberina isaky ny fitsingerenany ireo lefa trigonometrika fototra enina, dia misy sanda samihafa ampidirina aminy izay mitovy sanda avoaka, izany hoe tsy manana fifandraisana iray amin' ny iray izy ireo. Noho izany dia tsy manana lefa mifamadika aminy izy ireo. Azo atao anefa ny mampananana lefa mifamadika azy ireo, amin' ny famerana ny vondron' ny sanda azo ampidirina amin' ilay lefa trigonometrika.^[19]^48ff

Ity tabilao manaraka ity dia manome ny anaran' ireo lefa mifamadika amin' ny lefa trigonometrika, miaraka amin' ny vondron' ny sanda azo ampidirina sy ny vondron' ny sanda azo avoaka.^[19]^48ff^[20]^521ff Ny sanda fototra voafidy dia ilay sanda raisina mahazatra amin' ny lefa mifamadika, ao anatin' ny elanelana aseho eto.

Anarana	Fanoratana mahazatra	Famaritana	Sanda azon' ny x raisina mba ho isa tena ny valiny	Vondron' ny sanda fototra voafidy mahazatra (radiana)	Vondron' ny sanda fototra voafidy mahazatra (degre fandrefesana zoro)
arcsine	$y = \arcsin (x)$	$x = \sin (y)$	$- 1 \leq x \leq 1$	$- \frac{π}{2} \leq y \leq \frac{π}{2}$	$- 9 0^{\circ} \leq y \leq 9 0^{\circ}$
arccosine	$y = \arccos (x)$	$x = \cos (y)$	$- 1 \leq x \leq 1$	$0 \leq y \leq π$	$0^{\circ} \leq y \leq 18 0^{\circ}$
arctangent	$y = \arctan (x)$	$x = \tan (y)$	Isa tena rehetra	$- \frac{π}{2} < y < \frac{π}{2}$	$- 9 0^{\circ} < y < 9 0^{\circ}$
arccotangent	$y = arccot (x)$	$x = \cot (y)$	Isa tena rehetra	$0 < y < π$	$0^{\circ} < y < 18 0^{\circ}$
arcsecant	$y = arcsec (x)$	$x = \sec (y)$	$x \leq - 1$ na $1 \leq x$	$0 \leq y < \frac{π}{2}$ na $\frac{π}{2} < y \leq π$	$0^{\circ} \leq y < 9 0^{\circ}$ na $9 0^{\circ} < y \leq 18 0^{\circ}$
arccosecant	$y = arccsc (x)$	$x = \csc (y)$	$x \leq - 1$ na $1 \leq x$	$- \frac{π}{2} \leq y < 0$ na $0 < y \leq \frac{π}{2}$	$- 9 0^{\circ} \leq y < 0^{\circ}$ na $0^{\circ} < y \leq 9 0^{\circ}$

Fanehoana amin' ny fitambaran' ny andian-tsinga ahitana herin' ny miova

Rehefa raisina ho lefa amin' ny miova tena ireo taha trigonometrika, dia azo aseho amin' ny fitambaran' ny andian-tsinga Maclaurin izy ireo. Ohatra, toy izao manaraka izao ny fanehoana ny sine sy ny cosine.^[21]

\begin{matrix} \sin x & = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n + 1}}{(2 n + 1)!} \\ \cos x & = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots & = \sum_{n = 0}^{\infty} (- 1)^{n} \frac{x^{2 n}}{(2 n)!} \end{matrix}

Amin' ireo famaritana ireo dia azo faritana ho an' ny isa haro koa ny lefa trigonometrika.^[22] Rehefa itarina ho lefa amin' ny miova tena na haro izy ireo, dia mihatra amin' ny lefa eksponansialy amin' ny isa haro, izany hoe ny lefa manana endrika $e^{z}$ rehefa isa haro ny $z$ , ity raikipohy manaraka ity:

e^{x + i y} = e^{x} (\cos y + i \sin y) .

Tena ilaina io lefa eksponansialy amin' ny isa haro io rehefa soratana amin' ny alalan' ny lefa trigonometrika.^[23]^[24]

Fikajiana ny lefa trigonometrika

Anisan' ny fampiasana voalohany ny tabilao matematika ny lefa trigonometrika.^[25] Nampidirina tao amin' ny boky fianarana matematika ireny tabilao ireny, ary nampianarina hitady sanda ao anatiny ny mpianatra, sady nampianarina hanombana sanda eo anelanelan' ireo sanda voatanisa mba hahazoana valiny marina kokoa.^[26] Nisy maridrefy natokana ho an' ny lefa trigonometrika ny fitaovana fanaovana kajy misy tsorakazo mihetsika.^[27]

Manana bokotra hikajiana ireo lefa trigonometrika fototra ny milina kajy siantifika: $\sin$ , $\cos$ , $\tan$ , ary indraindray $cis$ , miaraka amin' ireo lefa mifamadika aminy.^[28] Ny ankamaroany dia ahafahana misafidy ny fomba fandrefesana zoro: degre, radiana, ary indraindray gradiana. Manolotra fitahirizana lefa efa voasoratra izay ahitana ny lefa trigonometrika ny ankamaroan' ny fiteny fanoratana rindrankajy amin' ny solosaina.^[29] Ny singa ara-pitaovana mikajy isa aseho amin' ny faingo mitsingevana, izay tafiditra ao amin' ireo singa elektronika madinika misy ny mikroprosesera ampiasain' ny ankamaroan' ny solosaina manokana, dia manana baiko efa voaorina ao anatiny hikajiana ny lefa trigonometrika.^[30]

Lefa trigonometrika hafa

Ankoatra ireo taha enina voatanisa teo aloha, dia misy lefa trigonometrika fanampiny izay nanana lanjany ara-tantara, na dia mahalana aza no ampiasaina ankehitriny. Anisan' izany ny lefa manome ny halavan' ny ampahan-kitsy mampifandray ireo teboka roa mamaritra ny zoro eo amin' ny faribolana manana halavan-tapaka iray avy amin' ny foibe (chord):

crd θ = 2 \sin \frac{θ}{2};

ny lefa versine, izay sanda azo rehefa esorina amin' ny iray ny cosine:

vers θ = 1 - \cos θ = 2 \sin^{2} \frac{θ}{2}

(efa hita tao amin' ireo tabilao voalohany izy io^[31]); ny lefa coversine, izay sanda azo rehefa esorina amin' ny iray ny sine:

covers θ = 1 - \sin θ = vers \frac{π}{2} - θ;

ny lefa haversine, izay antsasaky ny versine:

hav θ = \frac{1}{2} vers θ = \sin^{2} \frac{θ}{2};

^[32]

ary ny lefa exsecant, izay sanda azo rehefa analana iray ny secant:

exsec θ = \sec θ - 1.

Ny lisitry ny fitoviana trigonometrika dia manome fifandraisana hafa eo amin' ireo lefa ireo.

Fampiharana

Astronomia

Nandritra ny taonjato maro dia nampiasaina ny trigonometria amin' ny sfera mba hamaritana ny toeran' ny Masoandro sy ny Volana ary ny kintana,^[33] haminavina ny fanakona-masoandro sy ny fanakonam-bolana, ary hamaritana ny lalan' ny fihodinan' ny planeta.^[34]

Amin' ny vanim-potoana maoderina, ny fandrefesana amin' ny alalan' ny telozoro dia ampiasaina amin' ny astronomia mba handrefesana ny halavirana mankany amin' ny kintana akaiky,^[35] ary ampiasaina koa amin' ny rafitra famaritana toerana sy fitarihan-dalana amin' ny alalan' ny zanabolana.^[36]

Fitarihan-dalana

Tamin' ny tantaran' ny fitetezana an-dranomasina dia nampiasaina ny trigonometria mba hamaritana ny laharam-pehintanin' ny sambo sy ny halaviran-jorony atsinanana na andrefan' ny tsipika fototra mampitohy ny tendrontany roa, hanaovana sarin' ny lalana haleha, ary hanaovana kajy ny halavirana mandritra ny dia.^[37]

Mbola ampiasaina amin' ny fitarihan-dalana ny trigonometria, ohatra amin' ny GPS (rafitra famaritana toerana maneran-tany) sy ny faharanitan-tsaina voatr' olombelona ampiasaina amin' ny fiara mandeha tsy mila mpamily.^[38]

Fandrefesana tany

Amin' ny fandrefesana tany dia ampiasaina ny trigonometria hanaovana kajy ny halavana sy ny velarana ary ny zoro misy eo amin' ny zavatra samihafa.^[39]

Amin' ny ambaratonga midadasika kokoa, dia ampiasaina amin' ny jeografia ny trigonometria mba handrefesana ny halavirana eo anelanelan' ny toerana atao mari-toerana.^[40]

Lefa misy fitsingerenana

Ny lefa $s (x)$ (mena) dia fitambaran' ny lefa *sine* enina manana habetra samy hafa sy tahan' ny fiverenana samy hafa. Ny habetra dia ny elanelana ambony indrindra amin' ny sanda afovoany; ny tahan' ny fiverenana kosa dia ny isan' ny fiverenan' ny onja isaky ny venty fotoana. Ireo tahan' ny fiverenana ireo dia mifandray amin' ny maha-heny isa tsimivaky amin' ny tahan' ny fiverenan' ny onja fototra azy. Antsoina hoe andian-tsinga Fourier ny fitambaran' ireo lefa ireo. Ny fanovana Fourier, $S (f)$ (manga), izay mampiseho ny habetran' ny onja arakaraka ny tahan' ny fiverenany, dia mampiseho ireo tahan' ny fiverenan' ny onja enina (eo amin' ny heny tsiankasa amin' ny tahan' ny fiverenan' ny onja fototra) sy ny habetran' ireo onja (1/isa tsiankasa).

Fototry ny teôrian' ny lefa misy fitsingerenana, izany hoe lefa mamerina ny sandany isaky ny elanelana raikitra, ny lefa sine sy cosine,^[41] anisan' izany ireo lefa mamaritra ny onjan' ny feo sy ny onjam-pahazavana. Hitan' i Fourier fa ny lefa mitohy rehetra izay misy fitsingerenana dia azo faritana ho fitambarana tsy manam-petra misy lefa trigonometrika.

Na ny lefa tsy misy fitsingerenana aza dia azo aseho ho horona misy lefa sine sy cosine, amin' ny alalan' ny fanovana Fourier. Ny horona dia kajy manangona ny fandraisan' anjaran' ny sanda miova. Anisan' ny sehatra ampiharana izany ny mekanika kantika, izay mandinika ny fihetsiky ny zavatra amin' ny ambaratongan' ny atôma sy ny singa madinika kokoa,^[42] sy ny fifandraisan-davitra,^[43] ary misy sehatra hafa koa.

Siansa momba ny hazavana sy ny feo

Ampiasaina amin' ny siansa maro momba ny zavatra sy ny angovo ny trigonometria,^[44] anisan' izany ny siansa momba ny feo^[45] sy ny siansa momba ny hazavana.^[45] Amin' ireo sehatra ireo dia ampiasaina izy mba hamaritana ny onjan' ny feo sy ny onjam-pahazavana ary hamahana olana misy fepetra eny amin' ny sisiny sy olana momba ny fifindran' ny onja.^[46]

Fampiharana hafa

Anisan' ny sehatra hafa mampiasa trigonometria na lefa trigonometrika ny teôrian' ny mozika;^[47] ny fandrefesana ny endrika sy ny haben' ny Tany; ny famoronana feo amin' ny fitaovana elektronika;^[48] ny maritrano;^[49] ny elektronika;^[47] ny biolojia;^[50] ny fanaovana sary ara-pitsaboana, toy ny tomôdensitômetria (na fanaovana sary amin' ny alalan' ny taratra X avy amin' ny zoro maro, izay atambatry ny solosaina ho sarin' ny faritra ao anatin' ny vatana) sy ny ekôgrafia (na fanaovana sary amin' ny alalan' ny onjan' ny feo izay avo loatra ny tahan' ny fiverimberenany ka tsy ren' ny sofin' olombelona);^[51] ny simia;^[52] ny teôrian' ny isa, ary noho izany koa ny siansa momba ny fanakodiana sy ny famakiana hafatra miafina;^[53] ny siansa momba ny horohoron-tany;^[45] ny siansa momba ny atmosfera sy ny toetrandro;^[54] ny siansa momba ny ranomasina;^[55] ny fampihenana ny habetsahan' ny angona ilaina hitehirizana sary;^[56] ny fandinihana ny feon' ny fiteny;^[57] ny siansan' ny toekarena;^[58] ny injenieria momba ny herinaratra; ny injenieria momba ny milina; ny injenieria momba ny fotodrafitrasa;^[47] ny famoronana sy fanodinana sary amin' ny solosaina;^[59] ny fanaovana sarintany;^[47] ny fandinihana ny firafitry ny kristaly;^[60] ary ny famolavolana sy fanamboarana lalao video.^[59]

Fitoviana

Lahatsoratra amin' ny antsipiriany: lisitry ny fitoviana trigonometrika.

Anisan' ny mampiavaka ny trigonometria ny fisian' ny fitoviana maro, izany hoe fampimirana marina amin' ny sanda rehetra azo ampidirina.^[61]

Antsoina hoe fitoviana trigonometrika ny fitoviana izay zoro ihany no tafiditra ao aminy. Misy fampimirana hafa antsoina hoe fitoviana momba ny telozoro,^[62] izay mampifandray ny lafy sy ny zoro amin' ny telozoro iray.

Fitoviana momba ny telozoro

Amin' ireto fitoviana manaraka ireto, ny A, B ary C dia ny zoro amin' ny telozoro iray, ary ny a, b ary c dia ny halavan' ny lafy mifanatrika amin' ireo zoro ireo avy, araka ny asehon' ny kisary.

Fitsipika momba ny sine

Toy izao ny fitsipika momba ny sine (antsoina koa hoe « sine rule ») ho an' ny telozoro iray na inona na inona:^[63]

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R = \frac{a b c}{2 Δ},

izay ny $Δ$ no velaran' ny telozoro, ary ny R kosa no halavan' ny tapaka avy amin' ny foiben' ny faribolana mandalo amin' ny tendron' ny telozoro rehetra ka hatrany amin' io faribolana io:

R = \frac{a b c}{\sqrt{(a + b + c) (a - b + c) (a + b - c) (b + c - a)}} .

Fitsipika momba ny cosine

Ny fitsipika momba ny cosine (antsoina hoe raikipohy momba ny cosine, na « cos rule ») dia fanitarana ny teôrema momba ny telozoro misy zoro mahitsy (Pythagorean theorem) ho amin' ny telozoro rehetra:^[63]

c^{2} = a^{2} + b^{2} - 2 a b \cos C,

na, amin' ny endrika mitovy hevitra aminy:

\cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b} .

Fitsipika momba ny tangent

Ny fitsipika momba ny tangent, izay novolavolain' i François Viète, dia azo ampiasaina ho solon' ny fitsipika momba ny cosine rehefa tadiavina ny halavan' ny lafy mbola tsy fantatra amin' ny telozoro iray. Manamora ny kajy izany rehefa mampiasa tabilao misy ny sandan' ny lefa trigonometrika.^[64] Toy izao ilay fitsipika:

\frac{a - b}{a + b} = \frac{\tan [\frac{1}{2} (A - B)]}{\tan [\frac{1}{2} (A + B)]}

Velarana

Raha fantatra ny lafy roa a sy b ary ny zoro C eo anelanelan' ireo lafy ireo, dia azo amin' ny antsasaky ny vokatry ny fampitomboana ny halavan' ireo lafy roa amin' ny sine an' ilay zoro eo anelanelany ny velaran' ny telozoro:^[63]

Velarana = Δ = \frac{1}{2} a b \sin C

Fitoviana trigonometrika

Fitoviana mifandray amin' ny teôrema momba ny telozoro misy zoro mahitsy

Mifandray amin' ny teôrema momba ny telozoro misy zoro mahitsy ireto fitoviana trigonometrika manaraka ireto, ary marina amin' ny sanda rehetra:^[65]

\sin^{2} A + \cos^{2} A = 1

\tan^{2} A + 1 = \sec^{2} A

\cot^{2} A + 1 = \csc^{2} A

Ny fampimirana faharoa dia azo amin' ny fizarana ny fampimirana voalohany amin' ny $\cos^{2} A$ , ary ny fahatelo kosa amin' ny fizarana azy amin' ny $\sin^{2} A$ .

Raikipohin' i Euler

Ny raikipohin' i Euler, izay milaza fa $e^{i x} = \cos x + i \sin x$ , dia ahazoana ireto fitoviana manaraka ireto ao amin' ny fanadihadiana matematika, izay maneho ny sine, ny cosine ary ny tangent amin' ny alalan' ny e sy ny isa haro i, izay manana toradroa mitovy amin' ny −1:

\sin x = \frac{e^{i x} - e^{- i x}}{2 i}, \cos x = \frac{e^{i x} + e^{- i x}}{2}, \tan x = \frac{i (e^{- i x} - e^{i x})}{e^{i x} + e^{- i x}} .

Fitoviana trigonometrika hafa

Lahatsoratra amin' ny antsipiriany: lisitry ny fitoviana trigonometrika.

Anisan' ny fitoviana trigonometrika hafa fampiasa matetika ny fitoviana momba ny antsasan-joro, ny fitoviana momba ny fitambaran' ny zoro sy ny vokatry ny fanalana zoro iray amin' ny iray hafa, ary ny fitoviana manova vokatry ny fampitomboana ho fitambarana.^[8]

Tantara

Nandalina ny fandrefesana zoro ny astrônôma someriana, ka nampiasa ny fizarana ny faribolana ho degre 360.^[67] Izy ireo, ary tatỳ aoriana ny Babilôniana, dia nandalina ny tahan' ny halavan' ny lafin' ny telozoro mitovy endrika ary nahita toetra sasantsasany ananan' ireo taha ireo. Tsy novolavolainy ho fomba mirindra ahafahana mamantatra ny halavan' ny lafy sy ny refin' ny zoron' ny telozoro anefa izany. Nampiasa fomba mitovitovy amin' izany koa ny Nobiana fahiny.^[68]

Tamin' ny taonjato faha-3 talohan' i Kristy, ireo mpahay matematika tamin' ny vanim-potoana helenistika, toa an' i Euclid sy Archimedes, dia nandalina ny toetran' ny ampahan-kitsy mampifandray teboka roa eo amin' ny faribolana sy ny zoro voasoritra ao anatiny, izay manana ny tendrony eo amin' ilay faribolana ary ny lafiny manaraka ampahan-kitsy toy izany. Nanaporofo teôrema mitovy hevitra amin' ny raikipohy trigonometrika maoderina izy ireo, na dia tamin' ny fomba ara-jeômetria aza no nanehoany izany fa tsy tamin' ny aljebra. Tamin' ny taona 140 talohan' i Kristy, i Hipparchus, avy any Nicaea any Asia Minor, dia nanome ny tabilao voalohany mirakitra ny halavan' ireo ampahan-kitsy ireo, izay mitovitovy amin' ny tabilao maoderina mirakitra ny sandan' ny lefa sine. Nampiasainy hamahana olana momba ny trigonometria sy ny trigonometria amin' ny sfera izany.^[69] Tamin' ny taonjato faha-2 taorian' i Kristy, i Ptolemy, astrônôma grika-ejipsianina avy any Alexandria, Ejipta, dia nanamboatra tabilao trigonometrika amin' ny an-tsipiriany, dia ny tabilao mirakitra ny halavan' ireo ampahan-kitsy nataon' i Ptolemy, ao amin' ny boky 1, toko faha-11 amin' ny Almagest nosoratany.^[70] Nampiasa ny halavan' ny ampahan-kitsy mampifandray teboka roa eo amin' ny faribolana i Ptolemy mba hamaritana ny lefa trigonometrika nampiasainy. Fahasamihafana kely amin' ny fomba famaritana ny lefa sine ampiasaina ankehitriny izany.^[71] (Ny sanda antsoina hoe sin(θ) dia azo amin' ny fitadiavana ao amin' ny tabilaon' i Ptolemy ny halavan' ny ampahan-kitsy mifanaraka amin' ny zoro avo roa heny amin' ilay tadiavina, izany hoe 2θ, ary avy eo mizara roa an' io sanda io.) Taonjato maro no lasa vao nisy tabilao ahitana antsipiriany kokoa. Mbola nampiasaina hanaovana kajy trigonometrika amin' ny astrônômia kosa ny boky nosoratan' i Ptolemy nandritra ny 1200 taona nanaraka, tany amin' ny tontolo bizantina sy silamo tamin' ny Andro Antenatenany ary, tatỳ aoriana, tany Eorôpa Andrefana.

Ao amin' ny Surya Siddhanta no ahitana voalohany ny famaritana maoderina ny lefa sine. Mbola nofaritan' i Aryabhata, mpahay matematika sy astrônôma indiana, tamin' ny an-tsipiriany kokoa ny toetran' io lefa io tamin' ny taonjato faha-5 taorian' i Kristy.^[72] Nadika sy nitarin' ny mpahay matematika tao amin' ny tontolo silamo tamin' ny Andro Antenatenany ireo asa grika sy indiana ireo. Tamin' ny taona 830 taorian' i Kristy, i Habash al-Hasib al-Marwazi, mpahay matematika persianina, dia namorona ny tabilao voalohany mirakitra ny sandan' ny lefa cotangent.^[73]^[74] Tamin' ny taonjato faha-10 taorian' i Kristy, dia efa nampiasaina avokoa ny lefa trigonometrika enina tao amin' ny asan' i Abū al-Wafā' al-Būzjānī, mpahay matematika persianina.^[75] Nanana tabilao mirakitra ny sandan' ny lefa sine i Abu al-Wafa, ka 0.25° ny elanelan' ny zoro nifanesy ary nahatratra tarehimarika 8 aorian' ny faingo ny fahamarinan' ny sanda. Nanana tabilao marina mirakitra ny sandan' ny lefa tangent koa izy.^[76] Nanao fanavaozana lehibe tamin' ny trigonometria amin' ny sfera koa izy.^[77]^[78]^[36] I Nasir al-Din al-Tusi, manam-pahaizana persianina tamin' ny sehatra maro, dia voalaza fa namorona ny trigonometria ho taranja matematika mahaleo tena.^[79]^[80]^[81] Izy no voalohany nandray ny trigonometria ho taranja matematika tsy miankina amin' ny astrônômia, ary novolavolainy ho amin' ny endriny ankehitriny ny trigonometria amin' ny sfera.^[74] Notanisainy ireo tranga enina samy hafa momba ny telozoro misy zoro mahitsy amin' ny trigonometria amin' ny sfera. Tao amin' ny On the Sector Figure nosoratany, dia nambarany ny fitsipika momba ny lefa sine ho an' ny telozoro amin' ny lemaka sy amin' ny sfera, hitany ny fitsipika momba ny lefa tangent ho an' ny telozoro amin' ny sfera, ary nomeny ny fanaporofoana an' ireo fitsipika roa ireo.^[82] Tamin' ny taonjato faha-14, i Madhava of Sangamagrama dia namolavola ny fanehoana ny lefa trigonometrika sine, cosine ary arctangent ho fitambaran' ny andian-tsinga tsy manam-petra.^[83] Tsy voatahiry hatramin' izao ireo boky matematika nosoratan' i Madhava tany am-boalohany. Voatahiry ao amin' ny asan' ireo astrônôma tao Kerala tatỳ aoriana kosa ny fitambaran' ny andian-tsinga tsy manam-petra sy ny tabilao mirakitra ny sandan' ny lefa sine nataony, indrindra ao amin' ny Aryabhatiyabhasya sy ny Tantrasamgraha nosoratan' i Nilakantha Somayaji.^[84] Nanome fanaporofoana ara-jeômetria (upapatti) ho an' ireo fanehoana amin' ny alalan' ny fitambaran' ny andian-tsinga ireo i Nilakantha, ary nampiasa izany tamin' ny kajy astrônômika mifototra amin' ny sfera.^[85] Tonga tany Eorôpa Andrefana ny fahalalana momba ny lefa sy ny fomba trigonometrika tamin' ny alalan' ny fandikana ho amin' ny teny latina tamin' ny taonjato faha-12 ny Almagest nosoratan' i Ptolemy tamin' ny teny grika, ary koa ny asan' ireo astrônôma persianina sy arabo toa an' i Al Battani sy Nasir al-Din al-Tusi.^[86] Anisan' ny asa voalohany momba ny trigonometria nosoratan' ny mpahay matematika avy any Eorôpa Avaratra ny De Triangulis, nosoratan' i Regiomontanus, mpahay matematika alemàna tamin' ny taonjato faha-15. Nampirisika azy hanoratra izany sady nanome azy ny dika mitovy amin' ny Almagest ny kardinaly Basilios Bessarion, manam-pahaizana grika bizantina izay niarahany nipetraka nandritra ny taona maromaro.^[87] Tamin' izany fotoana izany koa, dia nahavita fandikana hafa ny Almagest avy amin' ny teny grika ho amin' ny teny latina i George of Trebizond, avy any Kreta.^[88] Mbola tsy dia fantatra tany Eorôpa Avaratra tamin' ny taonjato faha-16 ny trigonometria, hany ka nanokana toko roa tao amin' ny De revolutionibus orbium coelestium i Nicolaus Copernicus mba hanazavana ny foto-kevitra fototra momba azy.

Noho ny filàna amin' ny famaritana ny toerana sy ny lalana arahina mandritra ny dia, ary ny fitomboan' ny filàna sarintany marina momba ny faritra ara-jeôgrafia midadasika, dia nanjary sampana lehibe amin' ny matematika ny trigonometria.^[89] I Bartholomaeus Pitiscus no voalohany nampiasa ilay teny, tamin' ny namoahany ny Trigonometria tamin' ny 1595.^[90] I Gemma Frisius no nanazava voalohany ny fomba fandrefesana amin' ny alalan' ny telozoro, izay mbola ampiasaina amin' ny fandrefesana tany ankehitriny. I Leonhard Euler kosa no nampiditra tanteraka ny isa haro tao amin' ny trigonometria. Nisy fiantraikany tamin' ny fivoaran' ny fitambaran' ny andian-tsinga trigonometrika ny asan' ireo mpahay matematika skôtlandey, dia i James Gregory tamin' ny taonjato faha-17 sy i Colin Maclaurin tamin' ny taonjato faha-18.^[91] Tamin' ny taonjato faha-18 koa i Brook Taylor no namaritra ny endrika ankapoben' ny fitambaran' ny andian-tsinga Taylor.^[92]

Jereo koa

Tabilaon' ny lefa sine nataon' i Aryabhata
Trigonometria nitarina
Sferan' i Lénárt
Lisitry ny lohahevitra momba ny telozoro
Lisitry ny fitoviana trigonometrika
Trigonometria mifototra amin' ny aljebra (rational trigonometry)
Telozoro tena tery
Fanakaikezana ho an' ny zoro kely
Lefa trigonometrika
Faribolana izay 1 ny halavan' ny tapaka mampifandray ny foibeny amin' ny teboka eo aminy
Fampiasana ny trigonometria

Loharano

↑ Online Etymology Dictionary, « trigonometry ». Daty nitsidihana: 2022-03-18.
↑ R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002).
↑ Trigonometry | Definition, Formulas, Ratios, & Identities | Britannica. www.britannica.com, 2026-07-23. Amin' ny teny anglisy. Daty nitsidihana: 2026-08-27.
↑ Boyer (1991), A History of Mathematics. Tsy voatondro ny pejy.
↑ Charles William Hackley, A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables. G. P. Putnam, 1853.
↑ Mary Jane Sterling, Trigonometry For Dummies. John Wiley & Sons, 24 February 2014, p. 185. ISBN 978-1-118-82741-3.
↑ Ron Larson; Robert P. Hostetler, Trigonometry. Cengage Learning, 10 March 2006, p. 230. ISBN 0-618-64332-X.
1 2 James Stewart; Lothar Redlin; Saleem Watson. Algebra and Trigonometry. Cengage Learning, 16 January 2015, p. 448. ISBN 978-1-305-53703-3.
↑ Dick Jardine; Amy Shell-Gellasch. Mathematical Time Capsules: Historical Modules for the Mathematics Classroom. MAA, 2011, p. 182. ISBN 978-0-88385-984-1.
↑ Krystle Rose Forseth; Christopher Burger; Michelle Rose Gilman; Deborah J. Rumsey. Pre-Calculus For Dummies. John Wiley & Sons, 2008, p. 218. ISBN 978-0-470-16984-1.
↑ Weisstein, Eric W. SOHCAHTOA. MathWorld.
↑ Chris Humble. Key Maths : GCSE, Higher. Fiona McGill. Stanley Thornes Publishers, Cheltenham, 2001, p. 51. ISBN 0-7487-3396-5. OCLC 47985033.
↑ Fehezanteny mety kokoa ho an' ny sekoly ambaratonga faharoa ny hoe “Some Old Horse Came A'Hopping Through Our Alley” (izany hoe, amin' ny teny malagasy: nisy soavaly antitra nitsambikimbikina namakivaky ny lalan-kelinay). Jonathan K. Foster. Memory: A Very Short Introduction. Oxford, 2008, p. 128. ISBN 978-0-19-280675-8.
1 2 David Cohen; Lee B. Theodore; David Sklar. Precalculus: A Problems-Oriented Approach, Enhanced Edition. Cengage Learning, 17 July 2009. ISBN 978-1-4390-4460-5.
↑ W. Michael Kelley. The Complete Idiot's Guide to Calculus. Alpha Books, 2002, p. 45. ISBN 978-0-02-864365-6.
↑ Jenny Olive. Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students. Cambridge University Press, 18 September 2003, p. 175. ISBN 978-0-521-01707-7.
↑ Mary P Attenborough. Mathematics for Electrical Engineering and Computing. Elsevier, 30 June 2003, p. 418. ISBN 978-0-08-047340-6.
↑ Ron Larson; Bruce H. Edwards. Calculus of a Single Variable. Cengage Learning, 10 November 2008, p. 21. ISBN 978-0-547-20998-2.
1 2 Elizabeth G. Bremigan; Ralph J. Bremigan; John D. Lorch. Mathematics for Secondary School Teachers. MAA, 2011. ISBN 978-0-88385-773-1.
↑ Martin Brokate; Pammy Manchanda; Abul Hasan Siddiqi. Calculus for Scientists and Engineers. Springer, 3 August 2019. ISBN 9789811384646.
↑ Serge Lang. Complex Analysis. Springer, 14 March 2013, p. 63. ISBN 978-3-642-59273-7.
↑ Silvia Maria Alessio. Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications. Springer, 9 December 2015, p. 339. ISBN 978-3-319-25468-5.
↑ K. RAJA RAJESWARI; B. VISVESVARA RAO. SIGNALS AND SYSTEMS. PHI Learning, 24 March 2014, p. 263. ISBN 978-81-203-4941-4.
↑ John Stillwell. Mathematics and Its History. Springer Science & Business Media, 23 July 2010, p. 313. ISBN 978-1-4419-6053-5.
↑ Martin Campbell-Kelly; Mary Croarken; Raymond Flood; Eleanor Robson. The History of Mathematical Tables: From Sumer to Spreadsheets. OUP Oxford, 2 October 2003. ISBN 978-0-19-850841-0.
↑ George S. Donovan; Beverly Beyreuther Gimmestad. Trigonometry with calculators. Prindle, Weber & Schmidt, 1980. ISBN 978-0-87150-284-1.
↑ Ross Raymond Middlemiss. Instructions for Post-trig and Mannheim-trig Slide Rules. Frederick Post Company, 1945.
↑ Calculator keys—what they do. Popular Science, April 1974, Bonnier Corporation, p. 125.
↑ Steven S. Skiena; Miguel A. Revilla. Programming Challenges: The Programming Contest Training Manual. Springer Science & Business Media, 18 April 2006, p. 302. ISBN 978-0-387-22081-9.
↑ Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C. Intel, 2013.
↑ Carl B. Boyer (1991), A History of Mathematics, pp. xxiii–xxiv.
↑ Kaj L. Nielsen (1966), Logarithmic and Trigonometric Tables to Five Places, pp. xxiii–xxiv.
↑ Olinthus Gregory (1816). Elements of Plane and Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations. Baldwin, Cradock, and Joy.
↑ Otto Neugebauer (1948). Mathematical methods in ancient astronomy. Bulletin of the American Mathematical Society, 54 (11), p. 1013–1041. DOI: 10.1090/S0002-9904-1948-09089-9 (azo vakina malalaka).
↑ Michael Seeds; Dana Backman (5 January 2009). Astronomy: The Solar System and Beyond. Cengage Learning, p. 254. ISBN 978-0-495-56203-0.
1 2 Michael Willers (13 February 2018). Armchair Algebra: Everything You Need to Know From Integers To Equations. Book Sales, p. 37. ISBN 978-0-7858-3595-0.
↑ John Sabine (1800). The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc, p. 1.
↑ Mordechai Ben-Ari; Francesco Mondada (2018). Elements of Robotics. Springer, p. 16. ISBN 978-3-319-62533-1.
↑ George Roberts Perkins (1853). Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables. D. Appleton & Company.
↑ Charles W. J. Withers; Hayden Lorimer (14 December 2015). Geographers: Biobibliographical Studies. A&C Black, p. 6. ISBN 978-1-4411-0785-5.
↑ H. G. ter Morsche; J. C. van den Berg; E. M. van de Vrie (7 August 2003). Fourier and Laplace Transforms. Cambridge University Press, p. 61. ISBN 978-0-521-53441-3.
↑ Bernd Thaller (8 May 2007). Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena. Springer Science & Business Media, p. 15. ISBN 978-0-387-22770-2.
↑ M. Rahman (2011). Applications of Fourier Transforms to Generalized Functions. WIT Press. ISBN 978-1-84564-564-9.
↑ Lawrence Bornstein; Basic Systems, Inc (1966). Trigonometry for the Physical Sciences. Appleton-Century-Crofts.
1 2 3 John J. Schiller; Marie A. Wurster (1988). College Algebra and Trigonometry: Basics Through Precalculus. Scott, Foresman. ISBN 978-0-673-18393-4.
↑ Dudley H. Towne (5 May 2014). Wave Phenomena. Dover Publications. ISBN 978-0-486-14515-0.
1 2 3 4 E. Richard Heineman; J. Dalton Tarwater (1 November 1992). Plane Trigonometry. McGraw-Hill. ISBN 978-0-07-028187-5.
↑ Mark Kahrs; Karlheinz Brandenburg (18 April 2006). Applications of Digital Signal Processing to Audio and Acoustics. Springer Science & Business Media, p. 404. ISBN 978-0-306-47042-4.
↑ Kim Williams; Michael J. Ostwald (9 February 2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser, p. 260. ISBN 978-3-319-00137-1.
↑ Dan Foulder (15 July 2019). Essential Skills for GCSE Biology. Hodder Education, p. 78. ISBN 978-1-5104-6003-4.
↑ Luciano Beolchi; Michael H. Kuhn (1995). Medical Imaging: Analysis of Multimodality 2D/3D Images. IOS Press, p. 122. ISBN 978-90-5199-210-6.
↑ Marcus Frederick Charles Ladd (2014). Symmetry of Crystals and Molecules. Oxford University Press, p. 13. ISBN 978-0-19-967088-8.
↑ Gennady I. Arkhipov; Vladimir N. Chubarikov; Anatoly A. Karatsuba (22 August 2008). Trigonometric Sums in Number Theory and Analysis. Walter de Gruyter. ISBN 978-3-11-019798-3.
↑ (1943). Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943.
↑ Mary Sears; Daniel Merriman; Woods Hole Oceanographic Institution (1980). Oceanography, the past. Springer-Verlag. ISBN 978-0-387-90497-9.
↑ JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81). International Telecommunication Union, 1993. Nojerena tamin' ny 6 April 2019.
↑ Kirsten Malmkjaer (4 December 2009). The Routledge Linguistics Encyclopedia. Routledge, p. 1. ISBN 978-1-134-10371-3.
↑ Kamran Dadkhah (11 January 2011). Foundations of Mathematical and Computational Economics. Springer Science & Business Media, p. 46. ISBN 978-3-642-13748-8.
1 2 Christopher Griffith (12 November 2012). Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity (mila fisoratana anarana ny fidirana). CRC Press, p. 153. ISBN 978-1-136-13702-0.
↑ John Joseph Griffin (1841). A System of Crystallography, with Its Application to Mineralogy. R. Griffin, p. 119.
↑ Dugopolski (July 2002). Trigonometry I/E Sup. Addison Wesley. ISBN 978-0-201-78666-8.
↑ V&S EDITORIAL BOARD (6 January 2015). CONCISE DICTIONARY OF MATHEMATICS. V&S Publishers, p. 288. ISBN 978-93-5057-414-0.
1 2 3 Cynthia Y. Young (19 January 2010). Precalculus. John Wiley & Sons, p. 435. ISBN 978-0-471-75684-2.
↑ Ron Larson (29 January 2010). Trigonometry. Cengage Learning, p. 331. ISBN 978-1-4390-4907-5.
↑ John C. Peterson (2004). Technical Mathematics with Calculus (illustrated). Cengage Learning, p. 856. ISBN 978-0-7668-6189-3. Ampahany amin' ny pejy 856.
↑ Boyer (1991), "Greek Trigonometry and Mensuration", p. 162.
↑ Ric Pimentel; Terry Wall (2018). Cambridge IGCSE Core Mathematics. 4th edition. Hachette UK, p. 275. ISBN 978-1-5104-2058-8. Ampahany amin' ny pejy 275.
↑ Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag, p. 744. ISBN 978-3-540-06995-9.
↑ Thurston (1996), "Appendix 1: Hipparchus's Table of Chords", pp. 235–236.
↑ Toomer, G. (1998). Ptolemy's Almagest. Princeton University Press. Bibcode: 1998ptal.book.....T. ISBN 978-0-691-00260-6.
↑ Thurston (1996), "Appendix 3: Ptolemy's Table of Chords", pp. 239–243.
↑ Boyer (1991), p. 215.
↑ Jacques Sesiano (2000). Islamic mathematics. Ao amin' i Helaine Selin sy Ubiratan D'Ambrosio, mpamoaka lahatsoratra, Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media, p. 157. ISBN 978-1-4020-0260-1.
1 2 trigonometry. Encyclopædia Britannica. Nojerena tamin' ny 2008-07-21.
↑ Boyer (1991), p. 238.
↑ Boyer (1991), p. 238.
↑ Moussa, Ali (2011). Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations. Arabic Sciences and Philosophy 21 (1): 1–56. Cambridge University Press. DOI: 10.1017/S095742391000007X. S2CID: 171015175.
↑ Gingerich, Owen. Islamic astronomy. Scientific American 254.4 (1986): 74–83.
↑ Nasir al-Din al-Tusi. MacTutor History of Mathematics archive. Nojerena tamin' ny 2021-01-08. Teny nalaina: 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.
↑ Berggren, J. L. (October 2013). Islamic Mathematics. the cambridge history of science, vol. 2, pp. 62–83. Cambridge University Press. DOI: 10.1017/CHO9780511974007.004. ISBN 9780521594486.
↑ ṬUSI, NAṢIR-AL-DIN i. Biography. Encyclopaedia Iranica. Nojerena tamin' ny 2018-08-05. Teny nalaina: 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.
↑ J. Lennart Berggren (2007). Mathematics in Medieval Islam. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, p. 518. ISBN 978-0-691-11485-9.
↑ Charles Whish (1834). On the Hindu Quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala. Transactions of the Royal Asiatic Society of Great Britain and Ireland 3 (3): 509–523. DOI: 10.1017/S0950473700001221 (azo idirana malalaka). JSTOR: 25581775.
↑ Roy, Ranjan (1990). The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha. Mathematics Magazine 63 (5): 291–306. DOI: 10.1080/0025570X.1990.11977541.
↑ Divakaran, P. P. (2018). The Mathematics of India: Concepts, Methods, Connections. Springer Nature, pp. 281–289. DOI: 10.1007/978-981-13-1774-3. ISBN 978-981-13-1774-3.
↑ Boyer (1991), pp. 237, 274.
↑ Johann Müller Regiomontanus. MacTutor History of Mathematics archive. Nojerena tamin' ny 2021-01-08.
↑ N.G. Wilson (1992). From Byzantium to Italy. Greek Studies in the Italian Renaissance, London. ISBN 0-7156-2418-0.
↑ Ivor Grattan-Guinness (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 978-0-393-32030-5.
↑ Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance. Greenwood Publishing Group, p. 153. ISBN 978-0-313-32433-8.
↑ Ewald, William Bragg (2005-04-21). From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. OUP Oxford, p. 93. Amin' ny teny anglisy. ISBN 978-0-19-152309-0.
↑ Dempski, Kelly (November 2002). Focus on Curves and Surfaces. Premier Press, p. 29. Amin' ny teny anglisy. ISBN 978-1-59200-007-4.

Boky nampiasaina

Carl B. Boyer (1991). A History of Mathematics (fanontana Second; mila fisoratana anarana ny fidirana amin' ilay rohy). John Wiley & Sons, Inc. ISBN 978-0-471-54397-8.
Kaj L. Nielsen (1966). Logarithmic and Trigonometric Tables to Five Places (fanontana 2nd). New York: Barnes & Noble. LCCN 61-9103.
Hugh Thurston (1996). Early Astronomy. Springer Science & Business Media. ISBN 978-0-387-94822-5.

Vakiteny fanampiny

Trigonometric functions. Encyclopedia of Mathematics. EMS Press. 2001 [1994]. Famantarana: p/t094210.
Linton, Christopher M. (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge University Press.
Weisstein, Eric W. Trigonometric Addition Formulas. MathWorld. Anarana ao amin' ny adiresin-tranonkala: TrigonometricAdditionFormulas.

Rohy ivelany

Khan Academy (Akademia Khan): trigonometria, lesona fohy maimaim-poana amin' ny aterineto.
Trigonometry, nosoratan' i Alfred Monroe Kenyon sy Louis Ingold; The Macmillan Company, 1914. Aseho amin' ny endrika sary ny lahatsoratra manontolo.
Benjamin Banneker's Trigonometry Puzzle (ankamantatra momba ny trigonometria nataon' i Benjamin Banneker), ao amin' ny Convergence.
Dave's Short Course in Trigonometry (fampianarana fohy momba ny trigonometria), nosoratan' i David Joyce ao amin' ny Clark University (Oniversite Clark).
Trigonometry, nosoratan' i Michael Corral; miresaka ny fototry ny trigonometria ary zaraina araka ny GNU Free Documentation License — GNU FDL (fahazoan-dalana GNU ho an' ny tahirin-kevitra malalaka); rakitra PDF (endrika antontan-taratasy azo afindra).

[1] Online Etymology Dictionary, « trigonometry ». Daty nitsidihana: 2022-03-18.

[2] R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002).

[3] Trigonometry | Definition, Formulas, Ratios, & Identities | Britannica. www.britannica.com, 2026-07-23. Amin' ny teny anglisy. Daty nitsidihana: 2026-08-27.

[4] Boyer (1991), A History of Mathematics. Tsy voatondro ny pejy.

[Hackley1853-5] Charles William Hackley, A treatise on trigonometry, plane and spherical: with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables. G. P. Putnam, 1853.

[Sterling2014-6] Mary Jane Sterling, Trigonometry For Dummies. John Wiley & Sons, 24 February 2014, p. 185. ISBN 978-1-118-82741-3.

[LarsonHostetler2006-7] Ron Larson; Robert P. Hostetler, Trigonometry. Cengage Learning, 10 March 2006, p. 230. ISBN 0-618-64332-X.

[StewartRedlin2015-8] 1 2 James Stewart; Lothar Redlin; Saleem Watson. Algebra and Trigonometry. Cengage Learning, 16 January 2015, p. 448. ISBN 978-1-305-53703-3.

[JardineShell-Gellasch2011-9] Dick Jardine; Amy Shell-Gellasch. Mathematical Time Capsules: Historical Modules for the Mathematics Classroom. MAA, 2011, p. 182. ISBN 978-0-88385-984-1.

[ForsethBurger2008-10] Krystle Rose Forseth; Christopher Burger; Michelle Rose Gilman; Deborah J. Rumsey. Pre-Calculus For Dummies. John Wiley & Sons, 2008, p. 218. ISBN 978-0-470-16984-1.

[11] Weisstein, Eric W. SOHCAHTOA. MathWorld.

[12] Chris Humble. Key Maths : GCSE, Higher. Fiona McGill. Stanley Thornes Publishers, Cheltenham, 2001, p. 51. ISBN 0-7487-3396-5. OCLC 47985033.

[13] Fehezanteny mety kokoa ho an' ny sekoly ambaratonga faharoa ny hoe “Some Old Horse Came A'Hopping Through Our Alley” (izany hoe, amin' ny teny malagasy: nisy soavaly antitra nitsambikimbikina namakivaky ny lalan-kelinay). Jonathan K. Foster. Memory: A Very Short Introduction. Oxford, 2008, p. 128. ISBN 978-0-19-280675-8.

[CohenTheodore2009-14] 1 2 David Cohen; Lee B. Theodore; David Sklar. Precalculus: A Problems-Oriented Approach, Enhanced Edition. Cengage Learning, 17 July 2009. ISBN 978-1-4390-4460-5.

[Kelley2002-15] W. Michael Kelley. The Complete Idiot's Guide to Calculus. Alpha Books, 2002, p. 45. ISBN 978-0-02-864365-6.

[Olive2003-16] Jenny Olive. Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students. Cambridge University Press, 18 September 2003, p. 175. ISBN 978-0-521-01707-7.

[Attenborough2003-17] Mary P Attenborough. Mathematics for Electrical Engineering and Computing. Elsevier, 30 June 2003, p. 418. ISBN 978-0-08-047340-6.

[LarsonEdwards2008-18] Ron Larson; Bruce H. Edwards. Calculus of a Single Variable. Cengage Learning, 10 November 2008, p. 21. ISBN 978-0-547-20998-2.

[BremiganBremigan2011-19] 1 2 Elizabeth G. Bremigan; Ralph J. Bremigan; John D. Lorch. Mathematics for Secondary School Teachers. MAA, 2011. ISBN 978-0-88385-773-1.

[BrokateManchanda2019-20] Martin Brokate; Pammy Manchanda; Abul Hasan Siddiqi. Calculus for Scientists and Engineers. Springer, 3 August 2019. ISBN 9789811384646.

[Lang2013-21] Serge Lang. Complex Analysis. Springer, 14 March 2013, p. 63. ISBN 978-3-642-59273-7.

[Alessio2015-22] Silvia Maria Alessio. Digital Signal Processing and Spectral Analysis for Scientists: Concepts and Applications. Springer, 9 December 2015, p. 339. ISBN 978-3-319-25468-5.

[RAJESWARIRAO2014-23] K. RAJA RAJESWARI; B. VISVESVARA RAO. SIGNALS AND SYSTEMS. PHI Learning, 24 March 2014, p. 263. ISBN 978-81-203-4941-4.

[Stillwell2010-24] John Stillwell. Mathematics and Its History. Springer Science & Business Media, 23 July 2010, p. 313. ISBN 978-1-4419-6053-5.

[Campbell-KellyCampbell-Kelly2003-25] Martin Campbell-Kelly; Mary Croarken; Raymond Flood; Eleanor Robson. The History of Mathematical Tables: From Sumer to Spreadsheets. OUP Oxford, 2 October 2003. ISBN 978-0-19-850841-0.

[DonovanGimmestad1980-26] George S. Donovan; Beverly Beyreuther Gimmestad. Trigonometry with calculators. Prindle, Weber & Schmidt, 1980. ISBN 978-0-87150-284-1.

[Middlemiss1945-27] Ross Raymond Middlemiss. Instructions for Post-trig and Mannheim-trig Slide Rules. Frederick Post Company, 1945.

[28] Calculator keys—what they do. Popular Science, April 1974, Bonnier Corporation, p. 125.

[29] Steven S. Skiena; Miguel A. Revilla. Programming Challenges: The Programming Contest Training Manual. Springer Science & Business Media, 18 April 2006, p. 302. ISBN 978-0-387-22081-9.

[30] Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 3A, 3B and 3C. Intel, 2013.

[31] Carl B. Boyer (1991), A History of Mathematics, pp. xxiii–xxiv.

[32] Kaj L. Nielsen (1966), Logarithmic and Trigonometric Tables to Five Places, pp. xxiii–xxiv.

[Gregory1816-33] Olinthus Gregory (1816). Elements of Plane and Spherical Trigonometry: With Their Applications to Heights and Distances Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations. Baldwin, Cradock, and Joy.

[34] Otto Neugebauer (1948). Mathematical methods in ancient astronomy. Bulletin of the American Mathematical Society, 54 (11), p. 1013–1041. DOI: 10.1090/S0002-9904-1948-09089-9 (azo vakina malalaka).

[SeedsBackman2009-35] Michael Seeds; Dana Backman (5 January 2009). Astronomy: The Solar System and Beyond. Cengage Learning, p. 254. ISBN 978-0-495-56203-0.

[Willers2018-36] 1 2 Michael Willers (13 February 2018). Armchair Algebra: Everything You Need to Know From Integers To Equations. Book Sales, p. 37. ISBN 978-0-7858-3595-0.

[Sabine1800-37] John Sabine (1800). The Practical Mathematician, Containing Logarithms, Geometry, Trigonometry, Mensuration, Algebra, Navigation, Spherics and Natural Philosophy, Etc, p. 1.

[38] Mordechai Ben-Ari; Francesco Mondada (2018). Elements of Robotics. Springer, p. 16. ISBN 978-3-319-62533-1.

[Perkins1853-39] George Roberts Perkins (1853). Plane Trigonometry and Its Application to Mensuration and Land Surveying: Accompanied with All the Necessary Logarithmic and Trigonometric Tables. D. Appleton & Company.

[WithersLorimer2015-40] Charles W. J. Withers; Hayden Lorimer (14 December 2015). Geographers: Biobibliographical Studies. A&C Black, p. 6. ISBN 978-1-4411-0785-5.

[MorscheBerg2003-41] H. G. ter Morsche; J. C. van den Berg; E. M. van de Vrie (7 August 2003). Fourier and Laplace Transforms. Cambridge University Press, p. 61. ISBN 978-0-521-53441-3.

[Thaller2007-42] Bernd Thaller (8 May 2007). Visual Quantum Mechanics: Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena. Springer Science & Business Media, p. 15. ISBN 978-0-387-22770-2.

[Rahman2011-43] M. Rahman (2011). Applications of Fourier Transforms to Generalized Functions. WIT Press. ISBN 978-1-84564-564-9.

[BornsteinInc1966-44] Lawrence Bornstein; Basic Systems, Inc (1966). Trigonometry for the Physical Sciences. Appleton-Century-Crofts.

[SchillerWurster1988-45] 1 2 3 John J. Schiller; Marie A. Wurster (1988). College Algebra and Trigonometry: Basics Through Precalculus. Scott, Foresman. ISBN 978-0-673-18393-4.

[Towne2014-46] Dudley H. Towne (5 May 2014). Wave Phenomena. Dover Publications. ISBN 978-0-486-14515-0.

[HeinemanTarwater1992-47] 1 2 3 4 E. Richard Heineman; J. Dalton Tarwater (1 November 1992). Plane Trigonometry. McGraw-Hill. ISBN 978-0-07-028187-5.

[KahrsBrandenburg2006-48] Mark Kahrs; Karlheinz Brandenburg (18 April 2006). Applications of Digital Signal Processing to Audio and Acoustics. Springer Science & Business Media, p. 404. ISBN 978-0-306-47042-4.

[WilliamsOstwald2015-49] Kim Williams; Michael J. Ostwald (9 February 2015). Architecture and Mathematics from Antiquity to the Future: Volume I: Antiquity to the 1500s. Birkhäuser, p. 260. ISBN 978-3-319-00137-1.

[Foulder2019-50] Dan Foulder (15 July 2019). Essential Skills for GCSE Biology. Hodder Education, p. 78. ISBN 978-1-5104-6003-4.

[BeolchiKuhn1995-51] Luciano Beolchi; Michael H. Kuhn (1995). Medical Imaging: Analysis of Multimodality 2D/3D Images. IOS Press, p. 122. ISBN 978-90-5199-210-6.

[Ladd2014-52] Marcus Frederick Charles Ladd (2014). Symmetry of Crystals and Molecules. Oxford University Press, p. 13. ISBN 978-0-19-967088-8.

[ArkhipovChubarikov2008-53] Gennady I. Arkhipov; Vladimir N. Chubarikov; Anatoly A. Karatsuba (22 August 2008). Trigonometric Sums in Number Theory and Analysis. Walter de Gruyter. ISBN 978-3-11-019798-3.

[54] (1943). Study Guide for the Course in Meteorological Mathematics: Latest Revision, Feb. 1, 1943.

[SearsMerriman1980-55] Mary Sears; Daniel Merriman; Woods Hole Oceanographic Institution (1980). Oceanography, the past. Springer-Verlag. ISBN 978-0-387-90497-9.

[56] JPEG Standard (JPEG ISO/IEC 10918-1 ITU-T Recommendation T.81). International Telecommunication Union, 1993. Nojerena tamin' ny 6 April 2019.

[Malmkjaer2009-57] Kirsten Malmkjaer (4 December 2009). The Routledge Linguistics Encyclopedia. Routledge, p. 1. ISBN 978-1-134-10371-3.

[Dadkhah2011-58] Kamran Dadkhah (11 January 2011). Foundations of Mathematical and Computational Economics. Springer Science & Business Media, p. 46. ISBN 978-3-642-13748-8.

[Griffith2012-59] 1 2 Christopher Griffith (12 November 2012). Real-World Flash Game Development: How to Follow Best Practices AND Keep Your Sanity (mila fisoratana anarana ny fidirana). CRC Press, p. 153. ISBN 978-1-136-13702-0.

[Griffin1841-60] John Joseph Griffin (1841). A System of Crystallography, with Its Application to Mineralogy. R. Griffin, p. 119.

[Dugopolski2002-61] Dugopolski (July 2002). Trigonometry I/E Sup. Addison Wesley. ISBN 978-0-201-78666-8.

[BOARD2015-62] V&S EDITORIAL BOARD (6 January 2015). CONCISE DICTIONARY OF MATHEMATICS. V&S Publishers, p. 288. ISBN 978-93-5057-414-0.

[Young2010-63] 1 2 3 Cynthia Y. Young (19 January 2010). Precalculus. John Wiley & Sons, p. 435. ISBN 978-0-471-75684-2.

[Larson2010-64] Ron Larson (29 January 2010). Trigonometry. Cengage Learning, p. 331. ISBN 978-1-4390-4907-5.

[65] John C. Peterson (2004). Technical Mathematics with Calculus (illustrated). Cengage Learning, p. 856. ISBN 978-0-7668-6189-3. Ampahany amin' ny pejy 856.

[66] Boyer (1991), "Greek Trigonometry and Mensuration", p. 162.

[67] Ric Pimentel; Terry Wall (2018). Cambridge IGCSE Core Mathematics. 4th edition. Hachette UK, p. 275. ISBN 978-1-5104-2058-8. Ampahany amin' ny pejy 275.

[68] Otto Neugebauer (1975). A history of ancient mathematical astronomy. 1. Springer-Verlag, p. 744. ISBN 978-3-540-06995-9.

[69] Thurston (1996), "Appendix 1: Hipparchus's Table of Chords", pp. 235–236.

[toomer-70] Toomer, G. (1998). Ptolemy's Almagest. Princeton University Press. Bibcode: 1998ptal.book.....T. ISBN 978-0-691-00260-6.

[71] Thurston (1996), "Appendix 3: Ptolemy's Table of Chords", pp. 239–243.

[72] Boyer (1991), p. 215.

[Sesiano-73] Jacques Sesiano (2000). Islamic mathematics. Ao amin' i Helaine Selin sy Ubiratan D'Ambrosio, mpamoaka lahatsoratra, Mathematics Across Cultures: The History of Non-western Mathematics. Springer Science+Business Media, p. 157. ISBN 978-1-4020-0260-1.

[Britannica-74] 1 2 trigonometry. Encyclopædia Britannica. Nojerena tamin' ny 2008-07-21.

[75] Boyer (1991), p. 238.

[76] Boyer (1991), p. 238.

[musa-77] Moussa, Ali (2011). Mathematical Methods in Abū al-Wafāʾ's Almagest and the Qibla Determinations. Arabic Sciences and Philosophy 21 (1): 1–56. Cambridge University Press. DOI: 10.1017/S095742391000007X. S2CID: 171015175.

[78] Gingerich, Owen. Islamic astronomy. Scientific American 254.4 (1986): 74–83.

[79] Nasir al-Din al-Tusi. MacTutor History of Mathematics archive. Nojerena tamin' ny 2021-01-08. Teny nalaina: 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.

[80] Berggren, J. L. (October 2013). Islamic Mathematics. the cambridge history of science, vol. 2, pp. 62–83. Cambridge University Press. DOI: 10.1017/CHO9780511974007.004. ISBN 9780521594486.

[81] ṬUSI, NAṢIR-AL-DIN i. Biography. Encyclopaedia Iranica. Nojerena tamin' ny 2018-08-05. Teny nalaina: 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.

[82] J. Lennart Berggren (2007). Mathematics in Medieval Islam. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, p. 518. ISBN 978-0-691-11485-9.

[charles-83] Charles Whish (1834). On the Hindu Quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala. Transactions of the Royal Asiatic Society of Great Britain and Ireland 3 (3): 509–523. DOI: 10.1017/S0950473700001221 (azo idirana malalaka). JSTOR: 25581775.

[84] Roy, Ranjan (1990). The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha. Mathematics Magazine 63 (5): 291–306. DOI: 10.1080/0025570X.1990.11977541.

[85] Divakaran, P. P. (2018). The Mathematics of India: Concepts, Methods, Connections. Springer Nature, pp. 281–289. DOI: 10.1007/978-981-13-1774-3. ISBN 978-981-13-1774-3.

[86] Boyer (1991), pp. 237, 274.

[87] Johann Müller Regiomontanus. MacTutor History of Mathematics archive. Nojerena tamin' ny 2021-01-08.

[88] N.G. Wilson (1992). From Byzantium to Italy. Greek Studies in the Italian Renaissance, London. ISBN 0-7156-2418-0.

[89] Ivor Grattan-Guinness (1997). The Rainbow of Mathematics: A History of the Mathematical Sciences. W.W. Norton. ISBN 978-0-393-32030-5.

[90] Robert E. Krebs (2004). Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance. Greenwood Publishing Group, p. 153. ISBN 978-0-313-32433-8.

[91] Ewald, William Bragg (2005-04-21). From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics. OUP Oxford, p. 93. Amin' ny teny anglisy. ISBN 978-0-19-152309-0.

[92] Dempski, Kelly (November 2002). Focus on Curves and Surfaces. Premier Press, p. 29. Amin' ny teny anglisy. ISBN 978-1-59200-007-4.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[65]

[66]

[67]

[68]

[69]

[70]

[71]

[72]

[73]

[74]

[75]

[76]

[77]

[78]

[79]

[80]

[81]

[82]

[83]

[84]

[85]

[86]

[87]

[88]

[89]

[90]

[91]

[92]