# Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja
Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja sotir pakava ke solokseropa. Bata opa va skeda wale soluma is gentim koe barkrileem sosuler. Ina va barkrilekwa is barkrilotsa is barkrilidja dere sosuler.

## Atoera

### Izva va opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja

#### Taneaf brubeeem ta barkrilsabera
Xanta ke bata opa al sotir dene seltay ke savsavafa Misra is ke Mesopotamia is ke krant ke Nindhu Nadee bost, abdi voltol-decite. Sane ke KÁ.DINGIR.RA seltay va bata opa nuve rigavayar gu bolk ton tev-sane omava.
Plimpton 322 valpey ke KÁ.DINGIR.RA seltay ( mon -1800 ) va geaki ke opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja nuve atoer.

#### Ellasaf keltopikeem
Hipparque keltopik isu solokseropik ( -190 ; -120 ) va taneafa wafra icde bata opa vegeduyur kan yon wazdel : sin askiyid da kota voda ke gentim koe av vas  abrotce ke wazdel remazavzan koe ivamu kare lana titickafa olya vadjer. Bata patavara vas jontolafa barkrilekwa ke mialaf gentim sovadjer, trabe dacoba noelkeon yoltana gu wafra icde barkrilekwa. Neken, larde wafreem ke Hipparque solokseropik ten kruldeyer, pune bata wafra anton zo gruped vey Claúdios Ptolemaîos ellasik kiren in va sina sanegayar bak mon 150-e, do sinafe vegeduracave koe Mégistè grabom. To batinde wafreem gin zo gruper mali klaamialuk tuke Georg von Purbach keltopik is tuke Regiomontanus tavembik. Pu Menelaos ho Alexandreus ( tici 1-eafa decemda ) zo gar vonera icde pozafa opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja. Bata vonera icle pakavon tigir koe Mégistè grabom ise abrotcion zo gayar pu Claúdios Ptolemaîos keltopik.

#### Barataf solokseropikeem
Moni 400-e, Surya Siddhanta bioga gan baratik zo sutelayar. Ina tir icde keltopa ise va ellasafa keltopinda malgruper voxe icde opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja warzur. Solve va wazdelsabe gu tra ellasaf solokseropikeem gesiayar, tel suteks abdualbar va gesiara va wazdelacku gu kodafuyuna tra. Batcoba va stravok icde barkrilekwa di kruldesir. Azon, arabaf solokseropikeem milkane di vayar. Āryabhaṭa baratasolokseropik bak 499 va wafra dem barkrilekwa isu barkrilotsa dafuyur. In va barkrilekwa gu zya yoltayar ise va barkrilotsa gu kotizya ise va volaca ke barkrilekwa gu otkram zya.

Bak 628, ar barataf solokseropik, i Brahmagupta, va walef ueem malsaveyer patavatason va voda ke barkrilekwa kale toleafa vurinda.

#### Vankeda koe islamevaxo
Koe to islamevaxo bata opa, va ravalduga vas ugdacka urlickeyer. Ina batinde toz tiyir amidaf gu kocopa.

Abu Al-Wafa keltopik ( 940-998 ) va mégistè grabom ke Claúdios Ptolemaîos tuopelayar ikaplekuson va malsavera va gekast ke Claúdios Ptolemaîos ( dan gu nuva kan balemkril isu tevoya tokoda yoltayar ) ika tazukoy ke telafa opa gundaf gu tel minaf ( tula : barkrilekwa ke dokaks va 2 tra ). Omar Khayyām grupeik (1048-1131) va malsavera va bata opa do vanpokeucafa rietova zineyer dafutuson va nuva ta bralkara va miltay ke pataropa vey darkasaberopa. Pintakirafa nuva ta wafrara va barkrilekwa is va barkrilotsa icde kot gentim al zo suted gan Bhāskarācārya solokseropik bak 1150. In dere va pozkorafa opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja voneyer. Bak 13eafa decemda, Abū Jaʿfar Muḥammad ibn Muḥammad ibn al‐Ḥasan Naṣīr al‐Dīn al‐Ṭūsī solokseropik, kadimi Bhāskarācārya, ape tiyir riwe tel taneik gu torigira va opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja, wetce ugda solwifa gu solokseropa. Adim, bak 14eafa decemda, Ghiyath ad-Din Jamshid Mas`ud al-Kashi solokseropik, va fli ke bata opa wafrayar bal intafa vayara va keltopa.

#### Koe Europa: kosmara va Claúdios Ptolemaîos vulkunik
Bak 1220, koe Europa, Leonardo Fibonacci solokseropik va wafra ta sulera va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja kodrageyer ko Practica Geometriae neva neken bata neva va yona rokla aje doburer.

Inkera va elimafe sabe ko bata opa zo voneyer isti 15eafa decemda vey mallatinavara va grabomeem ke Claúdios Ptolemaîos. Icdef tanealik tid Georg von Peuerbach is moekote inaf vayasik i Regiomontanus. Va stravok ke barkrilekwa faveyen gan barataf solokseropik isu arabaf bat vayasik vanikatcuyur. Va wafra dem barkrilekwa kridjoveyer kan lya dem 600 000 lumki azu 10 000 000. Dere va wafra dem barkrilidja dafuyur. Bak 16eafa decemda, suteks ke Oronce Fine solokseropik isu Pedro Nunes isu Joachim Rheticus von Lauchen radimifiyid. Joachim va lya dem 1015 lumki do 10 ivamukiverast wafrayar. Bartholomäus Pitiscus solokseropik, bak 1595, va katcalaf kobaks icde opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja sanegayar.
Malsavera va lya disa va eka ke 10 wetce sabe is vonera va patavara dem sane omava teni 16eafa decemda, do François Viète is Simon Stevin, abicabicon jupayad da solokseropikeem va {\displaystyle \pi } ota wetce mancoba volsu kapbureda wal toloye abrotce.

### Rewara
Rewarinda va bata opa tid jontikapafa. Pilkovon, opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja zo faver koe kocopa isu totara vey moekote brube ke bargentim. Ar ind lize bata opa zo faver ( mepuskilafa vexala ) : altelopa is tralopa is poopa is mamopa is wiropa is enkopa is skapopa is bliropa is altopelopa is akolopa is sazopa is tawavopa is liwotopa is givavuropa is kot zavzaks.

## Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja
Fli ke bata opa zo rotentud kan barkril dem rontaf gentim.
Larde itaya va getimeem ke balkril sotir vas 180° ( ont π radiane ), pune tel lodavinkaf gentim dene man barkril sotir tel rontaf.
Koe bata ronefa vola, {\displaystyle \scriptstyle {\widehat {ACB}}} gentim tir tel rontaf. [AB] kril tir tel lodabrotcaf.
Fli ke bata opa cwade zo tentud, ton A ika {\displaystyle \scriptstyle {\widehat {BAC}}} gentim is ton sin ika barkrilekwa is ton cos ika barkrilotsa is ton tan ika barkrilidja :
{\displaystyle \sin A={{\mbox{tcakenaf kril}} \over {\mbox{tel lodabrotcaf kril}}}={a \over c}\qquad \cos A={{\mbox{pokeyaf kril}} \over {\mbox{tel lodabrotcaf kril}}}={b \over c}\qquad \tan A={{\mbox{tcakenaf kril}} \over {\mbox{pokeyaf kril}}}={a \over b}}
Bat tid tel lozolonaf tazukoy ke opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja. Al zo gotud ta gentimeem wal 0° is 90° ( ont wal 0 is π/2 radiane ). Malsaveson va ivamu wetce lum, va bat tentuks rodivatcet.

### Katcalafa milkuca
Kare bet A gentim, digit va ( sedme gekast ke Puthagóras ):
{\displaystyle \cos ^{2}A+\sin ^{2}A=1\,}

### Tazukoy ta dokaks isu baskaks ke ivamuki
Toloy nelkaf tazukoy tid tel dokas va barkrilekwa is barkrilotsa :
{\displaystyle \cos(A+B)=\cos A\cos B-\sin A\sin B} ;
{\displaystyle \sin(A+B)=\sin A\cos B+\cos A\sin B}.
Va tel ta barkrilidja maltrakut :
{\displaystyle \tan(A+B)={\frac {\tan A+\tan B}{1-\tan A\tan B}}}.
Dere va tazukoy mu baskaks maltrakut ( ikaplekuson va B gu B, grupeson da barkrilotsa fli tir tolokafi is da barkrilekwa fli isu barkrilidja tid voltolokafi ).

### Tazukoy ta jonkara va tra
{\displaystyle \cos(2a)=\cos ^{2}a-\sin ^{2}a=2\cos ^{2}a-1=1-2\sin ^{2}a\,}
{\displaystyle \sin(2a)=2\sin a\cos a\,}
{\displaystyle \tan(2a)={\frac {2\tan a}{1-\tan ^{2}a}}\,}
{\displaystyle \sin(3a)=3\sin a-4\sin ^{3}a\,}
{\displaystyle \cos(3a)=-3\cos a+4\cos ^{3}a\,}
{\displaystyle \tan(3a)={\frac {3\tan a-\tan ^{3}a}{1-3\tan ^{2}a}}}

### Tazukoy ta vonera isu volvonera (Tazukoy ke Simpson)
Dizveyeson va tazukoy ta dokaks is tel ta baskaks, maltrakut va :
Vonera
{\displaystyle \cos A\cos B={\frac {\cos(A-B)+\cos(A+B)}{2}}}, apton {\displaystyle \cos ^{2}A={\frac {1+\cos(2A)}{2}}},
{\displaystyle \sin A\sin B={\frac {\cos(A-B)-\cos(A+B)}{2}}}, apton {\displaystyle \sin ^{2}A={\frac {1-\cos(2A)}{2}}},
{\displaystyle \sin A\cos B={\frac {\sin(A+B)+\sin(A-B)}{2}}} ;
Volvonera
{\displaystyle \cos p+\cos q=2\cos {{p+q} \over 2}\cos {{p-q} \over 2}},
{\displaystyle \cos p-\cos q=-2\sin {{p+q} \over 2}\sin {{p-q} \over 2}},
{\displaystyle \sin p+\sin q=2\sin {{p+q} \over 2}\cos {{p-q} \over 2}}.

### Tazukoy ke tra vas miala
Bat tazukoy va jontik zvak vanpid. Abduplekuson va :
{\displaystyle t=\tan {\frac {A}{2}}},
Digit va :
{\displaystyle \sin A={\frac {2t}{1+t^{2}}}~,\qquad \cos A={\frac {1-t^{2}}{1+t^{2}}}~,\qquad \tan A={\frac {2t}{1-t^{2}}}}.

### Gekast ke Al-Kashi ok ke barkrilotsa
Benprostelanixeem minon batlize win, vas sabeeem ton saberopa ke Eὐkleίdhς kapburedar.
Ta ABC barkril ton krileem vas a = BC is b = AC is c = AB, digit va ( lova ke barkrilotsa ) :
{\displaystyle a^{2}=b^{2}+c^{2}-2\,b\,c\,\cdot \cos {\widehat {A}}}.
Bat tazukoy tir pilkovon zolonaf koe azentarura kan barkril ise koe keltopa, bak xanta, al zanudar. Tuke to al-Kashi solokseropik, ke Samarkand bema, gekast al zo turomalsavenar bak 15eafa decemda.
Katcalara : tode {\displaystyle {\hat {A}}=90^{\circ }} ik tode {\displaystyle \pi /2}, pune {\displaystyle a^{2}=b^{2}+c^{2}}, trabe gekast ke Puthagóras.

### Tenekara va barkril
Tenekara va barkril tir, larde tanoy kril is toloy pokeyaf gentim, ok 1 gentim is 2 pokeyaf kril, ok, vas nustuca, 2 kril i b is c is sinaf B gentim, trasira va vadjes barkril, trabe va a is b is c is A is B is C ( is ageltara va tana verta merewana koe abduaxa ).
Bata zvakinda kan tan abdif tazukoy zo maer ( do denaf momimas tazukoy : a = b · cos C + c · cos B).
Tula :
Bene Ox lya, OB = 1 isen OC = 1,5. OBM = 60° isen OCM = 30°. M zo gotrasir :
Turingat ; M tigir koe (x = 0,75 ; y = 0,45) monon.
Ovat : koe BMC barkril, B = 120° isen C = 30° numen M = 30° ; numen barkril tir dem toloy miltaf kril pokeyon gu B isen BM'= 0,5.
Azen {\displaystyle CM=2\times 0,5\times \cos(C)={{\sqrt {3}} \over 2}}.
Ont H momimaks ke M ben HM lya ; HM = y isen HMB gentim tir vas 30°.
{\displaystyle y={{\sqrt {3}} \over 4}\approx 0,43} isen {\displaystyle x=1-{1 \over 4}={3 \over 4}=0,75}.
Soluma ke {\displaystyle OM={{\sqrt {3}} \over 2}=MC} isen brog ke M tir vas 30° isen OMB gentim tir vas 90°.
Riafa da batcoba tir maneke opelafa isu vrendafa.
Jadion, wal balemoya ik tevoya satcafa woda zo erud. Patavasiko akenon va « kastara va barkril » al grisitad.
Sabera va eka ke tawavafa tra ke mialerpa ke Paris widava al zo skur siton gan Jean Picard kocopik isti 17eafa decemda.

### Welmot ke barkril
Tel A welwot ke barkril zo gotur kan abrotce ke 2 kril is ke barkrilekwa ke gentim sinon tazukan :
{\displaystyle A={\frac {1}{2}}a\times b\times \sin({\hat {C}})}.
Vey mana miltaca, rewana gu kot ontinuk ke barkril, va lova ke barkrilekwa romaltrakut.
Daref tazuk, tuvarana kan lova ke barkrilotsa, va exonera tazukoyack nover :
{\displaystyle A={\sqrt {p(p-a)(p-b)(p-c)}}},
lize a is b is c tid abrotceeem ke krileem isen p va anamezacku ke barkril dasugdar :
{\displaystyle p={(a+b+c) \over 2}}.

## Yona kotgrupafa funtexa

### Live is ruk ke ivamuki
- Larde live ke AB wazdel, va AB tra ikar do {\displaystyle {\widehat {AOB}}=2\alpha } : ont I rav ke AB, isen CD xakola remfisa va I ; pune do f = ID ok f = IC :
{\displaystyle f\cdot (2R-f)=(R\cdot \sin(\alpha ))^{2}}
- Welmot ke ruk :
{\displaystyle S=R^{2}\cdot \left[\alpha -{{\sin(2\alpha )} \over 2}\right]} ;
Viele α coba tir pinafa, bam va bat welmot gu tel ke zost dolunet {\displaystyle {2 \over 3}\cdot f\cdot AB} ( gekast ke Arkhimếdês ) : baskaks sotir kaike baroy.


### Tazukoy ke Machin
John Machin solokseropik tir taneik icde patavara va π do 100 woda kaike u, bak 1706, tuke intaf tazukoy. Man tazukoy ware zo faved noelkeon, patavatason va jontika woda kaike u dene π.

### Deraf jonkril
- San-perkril zo soredur kan gatcita is uwelt ( gekast ke Gauss-Wantzel ).
- Perkril is lerdkril zo someredud a patavara. Neken va sin roredut vey soara.
  - Wazdet da ta {\displaystyle A={{2\pi } \over 7}} ( miltay tigis koe redura va perkril vay soara ), digit va : {\displaystyle \sin(A)\cdot \sin(2A)\cdot \sin(3A)={{\sqrt {7}} \over 8}} , {\displaystyle \quad \cos(A)+\cos(2A)+\cos(3A)=-{1 \over 2}\quad } is {\displaystyle \quad \cos(A)\cdot \cos(2A)\cdot \cos(3A)={1 \over 8}}.
  - Gundaf tazukoy krulded ta lerdkril, tode {\displaystyle A={{2\pi } \over 9}} :{\displaystyle \sin(A)\cdot \sin(2A)\cdot \sin(4A)={{\sqrt {3}} \over 8}} , {\displaystyle \quad \cos(A)+\cos(2A)+\cos(4A)=0\quad } is {\displaystyle \quad \cos(A)\cdot \cos(2A)\cdot \cos(4A)={1 \over 8}}.


Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja
[[Ewava:Barkril dem A gentim.png|vignette|droite]]
[[Ewava:Unit circle angles color.svg|vignette|Ivamu ke bata opa is yon katcalaf gentim]]
[[Ewava:Table_of_Trigonometry,_Cyclopaedia,_Volume_2.jpg|vignette|droite|Zovdarak icde opa va skeda wal solum isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja, 1728 ''[[Cyclopaedia]]''.]]

Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja sotir pakava ke [[solokseropa]]. Bata opa va skeda wale soluma is gentim koe [[barkril]]eem sosuler. Ina va [[barkrilekwa]] is [[barkrilotsa]] is [[barkrilidja]] dere sosuler.

== Atoera ==
=== [[Izva va opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja]] ===
==== Taneaf brubeeem ta barkrilsabera ====
Xanta ke bata opa al sotir dene seltay ke [[savsavafa Misra]] is ke [[Mesopotamia]] is ke [[Seltay ke krant ke Nindhu Nadee bost|krant ke Nindhu Nadee bost]], abdi voltol-decite. Sane ke [[KÁ.DINGIR.RA]] seltay va bata opa nuve rigavayar gu bolk ton tev-sane omava.

[[Plimpton 322]] valpey ke KÁ.DINGIR.RA seltay (&nbsp;mon [[-1800]]&nbsp;) va geaki ke opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja nuve atoer.

==== Ellasaf keltopikeem ====
[[Hipparque]] keltopik isu solokseropik (&nbsp;[[-190]] ; [[-120]]&nbsp;) va taneafa wafra icde bata opa vegeduyur kan yon wazdel : sin askiyid da kota voda ke gentim koe av vas  abrotce ke wazdel remazavzan koe ivamu kare lana titickafa olya vadjer. Bata patavara vas jontolafa barkrilekwa ke mialaf gentim sovadjer, trabe dacoba noelkeon yoltana gu wafra icde barkrilekwa. Neken, larde wafreem ke Hipparque solokseropik ten kruldeyer, pune bata wafra anton zo gruped vey [[Claúdios Ptolemaîos]] ellasik kiren in va sina sanegayar bak mon 150<sup>-e</sup>, do sinafe vegeduracave koe ''[[Mégistè]]'' grabom. To batinde wafreem gin zo gruper mali klaamialuk tuke [[Georg von Purbach]] keltopik is tuke [[Regiomontanus]] tavembik. Pu [[Menelaos ho Alexandreus]] (&nbsp;tici 1<sup>-eafa</sup> decemda&nbsp;) zo gar vonera icde [[pozafa opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja]]. Bata vonera icle pakavon tigir koe Mégistè grabom ise abrotcion zo gayar pu Claúdios Ptolemaîos keltopik.

==== Barataf solokseropikeem ====
Moni [[400]]<sup>-e</sup>, [[Surya Siddhanta]] bioga gan baratik zo sutelayar. Ina tir icde keltopa ise va ellasafa keltopinda malgruper voxe icde opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja warzur. Solve va wazdelsabe gu tra ellasaf solokseropikeem gesiayar, tel suteks abdualbar va gesiara va wazdelacku gu kodafuyuna tra. Batcoba va stravok icde barkrilekwa di kruldesir. Azon, arabaf solokseropikeem milkane di vayar. [[Āryabhaṭa]] baratasolokseropik bak 499 va wafra dem barkrilekwa isu barkrilotsa dafuyur. In va barkrilekwa gu ''zya'' yoltayar ise va barkrilotsa gu ''kotizya'' ise va volaca ke barkrilekwa gu ''otkram zya''.<br>
Bak 628, ar barataf solokseropik, i [[Brahmagupta]], va walef ueem malsaveyer patavatason va voda ke barkrilekwa kale toleafa vurinda.

==== Vankeda koe islamevaxo ====
Koe to islamevaxo bata opa, va ravalduga vas ugdacka urlickeyer. Ina batinde toz tiyir amidaf gu kocopa.<br>
[[Abu Al-Wafa]] keltopik (&nbsp;[[940]]-[[998]]&nbsp;) va [[mégistè]] grabom ke Claúdios Ptolemaîos tuopelayar ikaplekuson va malsavera va [[gekast ke Claúdios Ptolemaîos]] (&nbsp;dan gu ''nuva kan balemkril isu tevoya tokoda'' yoltayar&nbsp;) ika tazukoy ke telafa opa gundaf gu tel minaf (&nbsp;tula : barkrilekwa ke dokaks va 2 tra&nbsp;). [[Omar Khayyām]] grupeik ([[1048]]-[[1131]]) va malsavera va bata opa do [[vanpokeucafa rietova]] zineyer dafutuson va nuva ta bralkara va miltay ke pataropa vey darkasaberopa. Pintakirafa nuva ta wafrara va barkrilekwa is va barkrilotsa icde kot gentim al zo suted gan [[Bhāskarācārya]] solokseropik bak 1150. In dere va [[pozkorafa opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja]] voneyer. Bak [[san-bareafa decemda|13<sup>eafa</sup> decemda]], [[Abū Jaʿfar Muḥammad ibn Muḥammad ibn al‐Ḥasan Naṣīr al‐Dīn al‐Ṭūsī]] solokseropik, kadimi Bhāskarācārya, ape tiyir riwe tel taneik gu torigira va opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja, wetce ugda solwifa gu solokseropa. Adim, bak [[san-balemeafa decemda|14<sup>eafa</sup> decemda]], [[Ghiyath ad-Din Jamshid Mas`ud al-Kashi]] solokseropik, va fli ke bata opa wafrayar bal intafa vayara va keltopa.

==== Koe Europa : kosmara va Claúdios Ptolemaîos vulkunik ====
Bak 1220, koe Europa, [[Leonardo Fibonacci]] solokseropik va wafra ta sulera va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja kodrageyer ko '''''Practica Geometriae''''' neva neken bata neva va yona rokla aje doburer.<br>
Inkera va elimafe sabe ko bata opa zo voneyer isti [[san-alubeafa decemda|15<sup>eafa</sup> decemda]] vey mallatinavara va grabomeem ke Claúdios Ptolemaîos. Icdef tanealik tid [[Georg von Peuerbach]] is moekote inaf vayasik i [[Regiomontanus]]. Va stravok ke barkrilekwa faveyen gan barataf solokseropik isu arabaf bat vayasik vanikatcuyur. Va wafra dem barkrilekwa kridjoveyer kan lya dem 600 000 lumki azu 10 000 000. Dere va wafra dem barkrilidja dafuyur. Bak [[san-teveafa decemda|16<sup>eafa</sup> decemda]], suteks ke [[Oronce Fine]] solokseropik isu [[Pedro Nunes]] isu [[Joachim Rheticus von Lauchen]] radimifiyid. Joachim va lya dem 10<sup>15</sup> lumki do 10 ivamukiverast wafrayar. [[Bartholomäus Pitiscus]] solokseropik, bak [[1595]], va katcalaf kobaks icde opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja sanegayar.

Malsavera va lya disa va eka ke 10 wetce sabe is vonera va patavara dem sane omava teni [[san-teveafa decemda|16<sup>eafa</sup> decemda]], do [[François Viète]] is [[Simon Stevin]], abicabicon jupayad da solokseropikeem va [[Pi|<math>\pi</math>]] ota wetce mancoba volsu kapbureda wal toloye abrotce.

=== Rewara ===
Rewarinda va bata opa tid jontikapafa. Pilkovon, opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja zo faver koe [[kocopa]] isu [[totara]] vey moekote brube ke [[bargentim]]. Ar ind lize bata opa zo faver (&nbsp;mepuskilafa vexala&nbsp;) : [[altelopa]] is [[tralopa]] is [[poopa]] is [[mamopa]] is [[wiropa]] is [[enkopa]] is [[skapopa]] is [[bliropa]] is [[altopelopa]] is [[akolopa]] is [[sazopa]] is [[tawavopa]] is [[liwotopa]] is [[givavuropa]] is kot zavzaks.

== Opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja ==
[[Ewava:Rtriangle.svg|vignette|droite|ABC barkril]]
Fli ke bata opa zo rotentud kan barkril dem rontaf gentim.

Larde itaya va getimeem ke balkril sotir vas 180° (&nbsp;ont π radiane&nbsp;), pune tel lodavinkaf gentim dene man barkril sotir tel rontaf.

Koe bata ronefa vola, <math>\scriptstyle\widehat{ACB}</math> gentim tir tel rontaf. [AB] kril tir tel lodabrotcaf.

Fli ke bata opa cwade zo tentud, ton '''''A''''' ika <math>\scriptstyle\widehat{BAC}</math> gentim is ton '''''sin''''' ika barkrilekwa is ton '''''cos''''' ika barkrilotsa is ton '''''tan''''' ika barkrilidja :

:<math> \sin A={\mbox{tcakenaf kril} \over \mbox{tel lodabrotcaf kril}} = {a \over c}
 \qquad \cos A={\mbox{pokeyaf kril} \over \mbox{tel lodabrotcaf kril}} = {b \over c}
 \qquad \tan A={\mbox{tcakenaf kril} \over \mbox{pokeyaf kril}} = {a \over b}
</math>

Bat tid tel lozolonaf tazukoy ke opa va skeda wal soluma isu gentim koe barkril is va barkrilekwa ik barkrilotsa ik barkrilidja. Al zo gotud ta gentimeem wal 0° is 90° (&nbsp;ont wal 0 is π/2 radiane&nbsp;). Malsaveson va ivamu wetce lum, va bat tentuks rodivatcet.

=== Katcalafa milkuca ===
Kare bet '''''A''''' gentim, digit va (&nbsp;sedme gekast ke Puthagóras&nbsp;):
: <math> \cos^{2} A + \sin^{2} A = 1 \,</math>

=== Tazukoy ta dokaks isu baskaks ke ivamuki ===
Toloy nelkaf tazukoy tid tel dokas va barkrilekwa is barkrilotsa :
:<math>\cos(A+B)=\cos A\cos B-\sin A\sin B</math> ;
:<math>\sin(A+B)=\sin A\cos B+\cos A\sin B</math>.
Va tel ta barkrilidja maltrakut :
:<math>\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}</math>.
Dere va tazukoy mu baskaks maltrakut (&nbsp;ikaplekuson va '''''B''''' gu '''''B''''', grupeson da barkrilotsa fli tir [[Tolokuca ke fli|tolokafi]] is da barkrilekwa fli isu barkrilidja tid [[Tolokuca ke fli|voltolokafi]]&nbsp;).

=== Tazukoy ta jonkara va tra ===
:<math>\cos (2a) =  \cos^{2} a - \sin^{2} a = 2\cos^{2} a - 1 = 1 - 2\sin^{2} a \,</math>

:<math>\sin (2a) =  2\sin a \cos a \,</math>

:<math>\tan (2a) = \frac{2\tan a}{1 - \tan^{2} a } \,</math>
:<math>\sin (3a) =  3\sin a -4 \sin^{3} a \,</math>

:<math>\cos (3a) =  -3\cos a +4 \cos^{3} a \,</math>
:<math>\tan (3a) =\frac{3\tan a-\tan^3a}{1-3\tan^2a}</math>

=== Tazukoy ta vonera isu volvonera (&nbsp;[[Tazukoy ke Simpson]]&nbsp;) ===
Dizveyeson va tazukoy ta dokaks is tel ta baskaks, maltrakut va :

;Vonera
:<math>\cos A \cos B=\frac{\cos(A-B)+\cos(A+B)}2</math>, apton <math>\cos^2A=\frac{1+\cos(2A)}2</math>,
:<math>\sin A \sin B=\frac{\cos(A-B)-\cos(A+B)}2</math>, apton <math>\sin^2A=\frac{1-\cos(2A)}2</math>,
:<math>\sin A \cos B=\frac{\sin(A+B)+\sin(A-B)}2</math> ;

;Volvonera
:<math>\cos p + \cos q = 2 \cos{{p+q} \over 2}\cos{{p-q} \over 2}</math>,
:<math>\cos p - \cos q = -2 \sin{{p+q} \over 2}\sin{{p-q} \over 2}</math>,
:<math>\sin p + \sin q = 2 \sin{{p+q} \over 2}\cos{{p-q} \over 2}</math>.

=== Tazukoy ke tra vas miala ===
Bat tazukoy va jontik zvak vanpid. Abduplekuson va :
:<math>t = \tan \frac A2</math>,
Digit va :
:<math>\sin A=\frac{2t}{1+t^2}~,\qquad\cos A=\frac{1-t^2}{1+t^2}~,\qquad\tan A=\frac{2t}{1-t^2}</math>.

=== Gekast ke Al-Kashi ok ke barkrilotsa ===
Benprostelanixeem minon batlize win, vas sabeeem ton [[saberopa ke Eὐkleίdhς]] kapburedar.
 
Ta ''ABC'' barkril ton krileem vas ''a = BC is b = AC'' is ''c = AB,'' digit va (&nbsp;[[lova ke barkrilotsa]]&nbsp;) :

:<math>a^2=b^2+c^2-2\, b \,c\, \cdot\cos\widehat A</math>.

Bat tazukoy tir pilkovon zolonaf koe azentarura kan barkril ise koe keltopa, bak xanta, al zanudar. Tuke to [[al-Kashi]] solokseropik, ke [[Samarkand]] bema, gekast al zo turomalsavenar bak [[san-alubeafa decemda|15<sup>eafa</sup> decemda]].

Katcalara : tode <math> \hat A = 90^\circ</math> ik tode <math>\pi/2</math>, pune <math>a^2 = b^2 + c^2</math>, trabe gekast ke Puthagóras.

=== Tenekara va barkril ===
Tenekara va barkril tir, larde tanoy kril is toloy pokeyaf gentim, ok 1 gentim is 2 pokeyaf kril, ok, vas nustuca, 2 kril i ''b'' is ''c'' is sinaf ''B'' gentim, trasira va vadjes barkril, trabe va ''a'' is ''b'' is ''c'' is ''A'' is ''B'' is ''C'' (&nbsp;is ageltara va tana verta merewana koe abduaxa&nbsp;).

Bata zvakinda kan tan abdif tazukoy zo maer (&nbsp;do denaf momimas tazukoy : ''a'' = ''b'' · cos ''C'' + ''c'' · cos ''B'').

Tula :
: Bene ''Ox'' lya, ''OB'' = 1 isen ''OC'' = 1,5. OBM = 60° isen OCM = 30°. M zo gotrasir :
: Turingat ; M tigir koe (''x'' = 0,75 ; ''y'' = 0,45) monon.
: Ovat : koe BMC barkril, ''B'' = 120° isen ''C'' = 30° numen ''M'' = 30° ; numen barkril tir dem toloy miltaf kril pokeyon gu ''B'' isen ''BM'''= 0,5.
: Azen <math>CM = 2 \times 0,5 \times \cos(C) = {\sqrt{3} \over 2}</math>.
: Ont ''H'' momimaks ke ''M'' ben ''HM'' lya ; ''HM'' = ''y'' isen HMB gentim tir vas 30°.
: <math>y = {\sqrt{3} \over 4} \approx 0,43</math> isen <math>x = 1 - {1 \over 4} = {3 \over 4} = 0,75</math>.
: Soluma ke <math>OM = {\sqrt{3} \over 2} = MC</math> isen brog ke ''M'' tir vas 30° isen OMB gentim tir vas 90°.
Riafa da batcoba tir maneke opelafa isu vrendafa.

Jadion, wal balemoya ik tevoya satcafa woda zo erud. Patavasiko akenon va «&nbsp;kastara va barkril&nbsp;» al grisitad.
Sabera va eka ke tawavafa [[tra ke mialerpa]] ke Paris widava al zo skur siton gan [[Jean Picard]] kocopik isti [[san-pereafa decemda|17<sup>eafa</sup> decemda]].

=== Welmot ke barkril ===
Tel A welwot ke barkril zo gotur kan abrotce ke 2 kril is ke barkrilekwa ke gentim sinon tazukan :
: <math>A =\frac12a \times b \times \sin(\hat C)</math>.
Vey mana miltaca, rewana gu kot ontinuk ke barkril, va [[lova ke barkrilekwa]] romaltrakut.

Daref tazuk, tuvarana kan [[lova ke barkrilotsa]], va exonera tazukoyack nover :
: <math>A = \sqrt{p(p-a)(p-b)(p-c)}</math>,
lize ''a'' is ''b'' is ''c'' tid abrotceeem ke krileem isen ''p'' va anamezacku ke barkril dasugdar :
: <math>p = {(a + b + c) \over 2}</math>.

== Yona kotgrupafa funtexa ==
=== Live is ruk ke ivamuki ===
* Larde live ke ''AB'' wazdel, va ''AB'' tra ikar do <math>\widehat{AOB} = 2 \alpha</math> : ont ''I'' rav ke ''AB'', isen ''CD'' xakola remfisa va ''I'' ; pune do ''f'' = ''ID'' ok ''f'' = ''IC'' :
*: <math> f \cdot (2R - f) = (R \cdot \sin(\alpha))^2</math>
* Welmot ke ruk :
*: <math>S = R^2 \cdot \left[\alpha - {{\sin(2\alpha)} \over 2}\right]</math> ;
*: Viele α coba tir pinafa, bam va bat welmot gu tel ke zost dolunet <math>{2 \over 3} \cdot f \cdot AB</math> (&nbsp;gekast ke Arkhimếdês&nbsp;) : baskaks sotir kaike baroy.

=== Tazukoy ke Machin ===
[[John Machin]] solokseropik tir taneik icde patavara va [[Pi|'''''π''''']] do 100 woda kaike u, bak 1706, tuke [[Tazukoy ke Machin|intaf tazukoy]]. Man tazukoy ware zo faved noelkeon, patavatason va jontika woda kaike u dene '''''π'''''.

=== Deraf jonkril ===
* San-perkril zo soredur kan gatcita is uwelt (&nbsp;[[gekast ke Gauss-Wantzel]]&nbsp;).
* Perkril is lerdkril zo someredud a patavara. Neken va sin roredut vey soara.
** Wazdet da ta <math>A = {{2\pi} \over 7}</math> (&nbsp;miltay tigis koe redura va perkril vay soara&nbsp;), digit va :<center> <math>\sin(A) \cdot \sin(2A) \cdot \sin(3A) = {\sqrt{7} \over 8}</math>&nbsp;, <math>\quad\cos(A) + \cos(2A) + \cos(3A) = - {1 \over 2}\quad</math> is <math>\quad\cos(A) \cdot \cos(2A) \cdot \cos(3A) = {1 \over 8}</math>.</center>
** Gundaf tazukoy krulded ta lerdkril, tode <math>A = {{2\pi} \over 9}</math> :<center><math>\sin(A) \cdot \sin(2A) \cdot \sin(4A) = {\sqrt{3} \over 8}</math>&nbsp;, <math>\quad\cos(A) + \cos(2A) + \cos(4A) = 0\quad</math> is <math>\quad\cos(A) \cdot \cos(2A) \cdot \cos(4A) = {1 \over 8}</math>.</center>

[[Loma:Solokseropa]]
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