Trigonometri chni fu nkciyi funkciyi kuta Voni mozhut buti viznacheni yak vidnoshennya dvoh storin ta kuta trikutnika abo yak vidnoshennya koordinat tochok kola Vidigrayut vazhlivu rol pri doslidzhenni periodichnih funkcij ta bagatoh ob yektiv Napriklad pri doslidzhenni ryadiv diferencialnih rivnyan Navedemo shist bazovih trigonometrichnih funkcij Ostanni chotiri viznachayutsya cherez pershi dvi Inshimi slovami voni ye oznachennyami a ne samostijnimi sutnostyami sinus sin a displaystyle sin alpha kosinus cos a displaystyle cos alpha tangens tg a sin a cos a displaystyle operatorname tg alpha tfrac sin alpha cos alpha kotangens ctg a cos a sin a displaystyle operatorname ctg alpha tfrac cos alpha sin alpha sekans sec a 1 cos a displaystyle sec alpha tfrac 1 cos alpha kosekans cosec a 1 sin a displaystyle operatorname cosec alpha tfrac 1 sin alpha Zmist 1 Oznachennya 1 1 Geometrichne viznachennya 2 Osnovni spivvidnoshennya 3 Teoremi dodavannya ta formuli dlya kratnih kutiv 3 1 Formuli dlya funkcij sumi kutiv 3 2 Formuli dlya funkcij podvijnih kutiv 3 3 Formuli dlya funkcij potrijnih kutiv 3 4 Formuli dlya funkcij polovinnih kutiv 3 5 Formuli dlya sumi funkcij kuta 3 6 Zagalni formuli dlya funkcij kratnih kutiv 4 Zagalni formuli dlya stepeniv funkcij 5 Rozkladi v ryad Tejlora 5 1 Zv yazok z eksponentoyu ta kompleksnimi chislami 6 Diferenciyuvannya ta integruvannya 7 Zv yazok z diferencialnim rivnyannyam 8 Vlastivosti ta zastosuvannya 8 1 Teorema sinusiv 8 2 Teorema kosinusiv 8 3 Teorema tangensiv 8 4 Teorema kotangensiv 8 5 Periodichni funkciyi 9 Div takozh 10 Primitki 11 Dzherela 12 PosilannyaOznachennya RedaguvatiGeometrichne viznachennya Redaguvati nbsp Viznachennya kutiv za dopomogoyu pryamokutnogo trikutnika nbsp Viznachennya trigonometrichnih funkcij na odinichnomu koli Trigonometrichni funkciyi mozhna viznachiti rozglyanuvshi pryamokutnij trikutnik Kosinusom kuta nazivayetsya vidnoshennya dovzhini prileglogo kateta do dovzhini gipotenuzi cos a A C A B b c cos b B C A B a c displaystyle cos alpha frac AC AB frac b c cos beta frac BC AB frac a c nbsp Sinusom kuta nazivayetsya vidnoshennya dovzhini protilezhnogo kateta do dovzhini gipotenuzi sin a B C A B a c sin b A C A B b c displaystyle sin alpha frac BC AB frac a c sin beta frac AC AB frac b c nbsp Tangensom kuta nazivayetsya vidnoshennya dovzhini protilezhnogo kateta do dovzhini prileglogo kateta tg a B C A C a b tg b A C B C b a displaystyle mbox tg alpha frac BC AC frac a b mbox tg beta frac AC BC frac b a nbsp Kotangensom kuta nazivayetsya vidnoshennya dovzhini prileglogo kateta do dovzhini protilezhnogo kateta ctg a A C B C b a ctg b B C A C a b displaystyle mbox ctg alpha frac AC BC frac b a mbox ctg beta frac BC AC frac a b nbsp Analogichnim chinom mozhna viznachiti trigonometrichni funkciyi na koli z odinichnim radiusom nbsp Odin period funkcij sin x displaystyle sin x nbsp ta cos x displaystyle cos x nbsp sin x displaystyle sin x nbsp ta cos x displaystyle cos x nbsp ce periodichni funkciyi iz periodom 2 p displaystyle 2 pi nbsp tg x displaystyle operatorname tg x nbsp ta ctg x displaystyle operatorname ctg x nbsp mayut period p displaystyle pi nbsp Spivvidnoshennya navedeni nizhche dozvolyayut viraziti znachennya trigonometrichnih funkcij vid dovilnogo dijsnogo argumentu cherez znachennya funkcij dlya argumentu z intervalu 0 p 2 displaystyle 0 pi over 2 nbsp sin x cos p 2 x displaystyle sin x cos left pi over 2 x right nbsp cos x sin p 2 x displaystyle cos x sin left pi over 2 x right nbsp tg x ctg p 2 x displaystyle operatorname tg x operatorname ctg left pi over 2 x right nbsp ctg x tg p 2 x displaystyle operatorname ctg x operatorname tg left pi over 2 x right nbsp Osnovni spivvidnoshennya RedaguvatiDokladnishe Spisok trigonometrichnih totozhnostej nbsp Nastupne spivvidnoshennya viplivaye iz teoremi Pifagora sin 2 x cos 2 x 1 displaystyle sin 2 x cos 2 x 1 nbsp Teoremi dodavannya ta formuli dlya kratnih kutiv RedaguvatiFormuli dlya funkcij sumi kutiv Redaguvati Z osnovnogo spivvidnoshennya sin a b sin a cos b cos a sin b displaystyle sin left alpha beta right sin alpha cos beta cos alpha sin beta nbsp otrimuyemo sin a b sin a cos b cos a sin b displaystyle sin left alpha pm beta right sin alpha cos beta pm cos alpha sin beta nbsp cos a b cos a cos b sin a sin b displaystyle cos left alpha pm beta right cos alpha cos beta mp sin alpha sin beta nbsp tg a b tg a tg b 1 tg a tg b ctg a b ctg a ctg b 1 ctg b ctg a displaystyle operatorname tg left alpha pm beta right operatorname tg alpha pm operatorname tg beta over 1 mp operatorname tg alpha operatorname tg beta operatorname ctg left alpha pm beta right operatorname ctg alpha operatorname ctg beta mp 1 over operatorname ctg beta pm operatorname ctg alpha nbsp Formuli dlya funkcij podvijnih kutiv Redaguvati sin 2 a 2 sin a cos a displaystyle sin 2 alpha 2 sin alpha cos alpha nbsp cos 2 a cos 2 a sin 2 a 2 cos 2 a 1 1 2 sin 2 a displaystyle cos 2 alpha cos 2 alpha sin 2 alpha 2 cos 2 alpha 1 1 2 sin 2 alpha nbsp tg 2 a 2 tg a 1 tg 2 a ctg 2 a ctg 2 a 1 2 ctg a 1 2 ctg a tg a displaystyle operatorname tg 2 alpha 2 operatorname tg alpha over 1 operatorname tg 2 alpha operatorname ctg 2 alpha operatorname ctg 2 alpha 1 over 2 operatorname ctg alpha 1 over 2 left operatorname ctg alpha operatorname tg alpha right nbsp Formuli dlya funkcij potrijnih kutiv Redaguvati sin 3 a 3 sin a 4 sin 3 a cos 3 a 4 cos 3 a 3 cos a displaystyle sin 3 alpha 3 sin alpha 4 sin 3 alpha cos 3 alpha 4 cos 3 alpha 3 cos alpha nbsp Formuli dlya funkcij polovinnih kutiv Redaguvati sin a 2 1 cos a 2 cos a 2 1 cos a 2 displaystyle sin alpha over 2 sqrt 1 cos alpha over 2 cos alpha over 2 sqrt 1 cos alpha over 2 nbsp tg a 2 sin a 1 cos a 1 cos a sin a ctg a 2 sin a 1 cos a 1 cos a sin a displaystyle operatorname tg alpha over 2 sin alpha over 1 cos alpha 1 cos alpha over sin alpha operatorname ctg alpha over 2 sin alpha over 1 cos alpha 1 cos alpha over sin alpha nbsp Formuli dlya sumi funkcij kuta Redaguvati a sin A b cos A r sin A B r cos p 2 A B a 2 b 2 sin A arctg b a r a 2 b 2 t g B b a displaystyle a sin A b cos A r sin left A B right r cos left pi over 2 A B right sqrt a 2 b 2 sin left A operatorname arctg b over a right r sqrt a 2 b 2 tgB b over a nbsp sin A sin B 2 sin A B 2 cos A B 2 displaystyle sin A pm sin B 2 sin A pm B over 2 cos A mp B over 2 nbsp cos A cos B 2 cos A B 2 cos A B 2 displaystyle cos A cos B 2 cos A B over 2 cos A B over 2 nbsp cos A cos B 2 sin A B 2 sin A B 2 displaystyle cos A cos B 2 sin A B over 2 sin A B over 2 nbsp tg A tg B sin A B cos A cos B ctg A ctg B sin B A sin A sin B displaystyle operatorname tg A pm operatorname tg B sin A pm B over cos A cos B operatorname ctg A pm operatorname ctg B sin B pm A over sin A sin B nbsp Formula dlya sumi bud yakoyi kilkosti sinusiv kutiv iz yih zsuvom i otrimannya odniyeyi funkciyi kuta A sin x a B sin x b C sin x g Y sin x Z cos x Y 2 Z 2 sin x arctg Z Y Y A cos a B cos b C cos g Z A sin a B sin b C sin g displaystyle A sin x alpha B sin x beta C sin x gamma Y sin x Z cos x sqrt Y 2 Z 2 sin x operatorname arctg Z over Y Y A cos alpha B cos beta C cos gamma Z A sin alpha B sin beta C sin gamma nbsp Zagalni formuli dlya funkcij kratnih kutiv Redaguvati Yaksho n ye cilim dodatnim chislom to sin n A n 1 cos n 1 A sin A n 3 cos n 3 A sin 3 A n 5 cos n 5 A sin 5 A displaystyle sin nA n choose 1 cos n 1 A sin A n choose 3 cos n 3 A sin 3 A n choose 5 cos n 5 A sin 5 A mp cdots nbsp cos n A cos n A n 2 cos n 2 A sin 2 A n 4 cos n 4 A sin 4 A displaystyle cos nA cos n A n choose 2 cos n 2 A sin 2 A n choose 4 cos n 4 A sin 4 A mp cdots nbsp Zagalni formuli dlya stepeniv funkcij RedaguvatiYaksho n ye cilim neparnim chislom to sin n x 1 n 1 2 2 n 1 sin n x n 1 sin n 2 x n 2 sin n 4 x n 3 sin n 6 x 1 n 1 2 n n 1 2 sin x displaystyle sin n x 1 n 1 over 2 over 2 n 1 left sin nx n choose 1 sin n 2 x n choose 2 sin n 4 x n choose 3 sin n 6 x cdots 1 n 1 over 2 n choose n 1 over 2 sin x right nbsp cos n x 1 2 n 1 cos n x n 1 cos n 2 x n 2 cos n 4 x n 3 cos n 6 x n n 1 2 cos x displaystyle cos n x left 1 over 2 right n 1 left cos nx n choose 1 cos n 2 x n choose 2 cos n 4 x n choose 3 cos n 6 x cdots n choose n 1 over 2 cos x right nbsp Yaksho n ye cilim parnim chislom to sin n x 1 n 2 2 n 1 cos n x n 1 cos n 2 x n 2 cos n 4 x n 3 cos n 6 x 1 n 2 2 n n 2 2 cos 2 x 1 2 n n n 2 displaystyle sin n x left 1 right n over 2 over 2 n 1 left cos nx n choose 1 cos n 2 x n choose 2 cos n 4 x n choose 3 cos n 6 x cdots left 1 right n 2 over 2 n choose n 2 over 2 cos 2x right 1 over 2 n n choose n over 2 nbsp cos n x 1 2 n 1 cos n x n 1 cos n 2 x n 2 cos n 4 x n 3 cos n 6 x n n 2 2 cos 2 x 1 2 n n n 2 displaystyle cos n x left 1 over 2 right n 1 left cos nx n choose 1 cos n 2 x n choose 2 cos n 4 x n choose 3 cos n 6 x cdots n choose n 2 over 2 cos 2x right 1 over 2 n n choose n over 2 nbsp Rozkladi v ryad Tejlora RedaguvatiIsnuyut taki rozkladi v ryad Tejlora trigonometrichnih funkcij sin x x x 3 3 x 5 5 x 7 7 n 0 1 n x 2 n 1 2 n 1 displaystyle sin x x frac x 3 3 frac x 5 5 frac x 7 7 cdots sum n 0 infty frac 1 n x 2n 1 2n 1 nbsp cos x 1 x 2 2 x 4 4 x 6 6 n 0 1 n x 2 n 2 n displaystyle cos x 1 frac x 2 2 frac x 4 4 frac x 6 6 cdots sum n 0 infty frac 1 n x 2n 2n nbsp tg x n 0 U 2 n 1 x 2 n 1 2 n 1 n 1 1 n 1 2 2 n 2 2 n 1 B 2 n x 2 n 1 2 n x x 3 3 2 x 5 15 17 x 7 315 62 x 9 2835 pri x lt p 2 displaystyle begin aligned operatorname tg x amp sum n 0 infty frac U 2n 1 x 2n 1 2n 1 amp sum n 1 infty frac 1 n 1 2 2n 2 2n 1 B 2n x 2n 1 2n amp x frac x 3 3 frac 2x 5 15 frac 17x 7 315 frac 62x 9 2835 cdots qquad text pri x lt frac pi 2 end aligned nbsp de U n displaystyle U n nbsp n te peretvorennya Bustrofedona B n displaystyle B n nbsp chisla Bernulli ta E n displaystyle E n nbsp chisla Ejlera cosec x n 0 1 n 1 2 2 2 n 1 1 B 2 n x 2 n 1 2 n 1 x x 6 7 x 3 360 31 x 5 15120 pri 0 lt x lt p displaystyle begin aligned operatorname cosec x amp sum n 0 infty frac 1 n 1 2 2 2n 1 1 B 2n x 2n 1 2n amp frac 1 x frac x 6 frac 7x 3 360 frac 31x 5 15120 cdots qquad text pri 0 lt x lt pi end aligned nbsp sec x n 0 U 2 n x 2 n 2 n n 0 1 n E 2 n x 2 n 2 n 1 x 2 2 5 x 4 24 61 x 6 720 pri x lt p 2 displaystyle begin aligned sec x amp sum n 0 infty frac U 2n x 2n 2n sum n 0 infty frac 1 n E 2n x 2n 2n amp 1 frac x 2 2 frac 5x 4 24 frac 61x 6 720 cdots qquad text pri x lt frac pi 2 end aligned nbsp ctg x n 0 1 n 2 2 n B 2 n x 2 n 1 2 n 1 x x 3 x 3 45 2 x 5 945 pri 0 lt x lt p displaystyle begin aligned operatorname ctg x amp sum n 0 infty frac 1 n 2 2n B 2n x 2n 1 2n amp frac 1 x frac x 3 frac x 3 45 frac 2x 5 945 cdots qquad text pri 0 lt x lt pi end aligned nbsp Zv yazok z eksponentoyu ta kompleksnimi chislami Redaguvati Vikoristovuyuchi vishenavedeni rozkladi v ryadi Tejlora mozhna pokazati sho funkciyi sin displaystyle sin nbsp ta cos displaystyle cos nbsp ye uyavnoyu ta dijsnoyu chastinami eksponenti chisto uyavnogo chisla e i 8 cos 8 i sin 8 displaystyle e i theta cos theta i sin theta nbsp Ce spivvidnoshennya nazivayetsya formuloyu Ejlera Mozhna viznachiti trigonometrichni funkciyi kompleksnoyi zminnoyi z sin z n 0 1 n 2 n 1 z 2 n 1 e i z e i z 2 i i sh i z displaystyle sin z sum n 0 infty frac 1 n 2n 1 z 2n 1 e iz e iz over 2i i operatorname sh left iz right nbsp cos z n 0 1 n 2 n z 2 n e i z e i z 2 ch i z displaystyle cos z sum n 0 infty frac 1 n 2n z 2n e iz e iz over 2 operatorname ch left iz right nbsp de i 2 1 displaystyle i 2 1 nbsp a sh x displaystyle operatorname sh x nbsp ta ch x displaystyle operatorname ch x nbsp vidpovidno giperbolichni sinus ta kosinus Dlya dijsnogo x displaystyle x nbsp mayut misce spivvidnoshennya cos x Re e i x sin x Im e i x displaystyle cos x operatorname Re e ix sin x operatorname Im e ix nbsp nbsp Kompleksnij sinus nbsp Kompleksnij kosinus nbsp Kompleksnij tangensDiferenciyuvannya ta integruvannya Redaguvati f x displaystyle f x nbsp d d x f x displaystyle frac d dx f x nbsp f x d x displaystyle int f x dx nbsp sin x displaystyle sin x nbsp cos x displaystyle cos x nbsp cos x C displaystyle cos x C nbsp cos x displaystyle cos x nbsp sin x displaystyle sin x nbsp sin x C displaystyle sin x C nbsp tg x displaystyle operatorname tg x nbsp sec 2 x displaystyle sec 2 x nbsp ln cos x C displaystyle ln left cos x right C nbsp ctg x displaystyle operatorname ctg x nbsp cosec 2 x displaystyle operatorname cosec 2 x nbsp ln sin x C displaystyle ln left sin x right C nbsp sec x displaystyle sec x nbsp sec x tg x displaystyle sec x operatorname tg x nbsp ln sec x tg x C displaystyle ln left sec x operatorname tg x right C nbsp cosec x displaystyle operatorname cosec x nbsp cosec x ctg x displaystyle operatorname cosec x operatorname ctg x nbsp ln cosec x ctg x C displaystyle ln left operatorname cosec x operatorname ctg x right C nbsp Zv yazok z diferencialnim rivnyannyam RedaguvatiFunkciyi sin x displaystyle sin x nbsp ta cos x displaystyle cos x nbsp ye rozv yazkami diferencialnogo rivnyannya garmonichnih kolivan d 2 y d x 2 y 0 displaystyle d 2 y over d x 2 y 0 nbsp Vlastivosti ta zastosuvannya RedaguvatiTeorema sinusiv Redaguvati Teorema sinusiv stverdzhuye sho dlya dovilnogo trikutnika zi storonami a displaystyle a nbsp b displaystyle b nbsp i c displaystyle c nbsp ta kutami sho protilezhni tim storonam A displaystyle A nbsp B displaystyle B nbsp i C displaystyle C nbsp sin A a sin B b sin C c 2 D a b c displaystyle frac sin A a frac sin B b frac sin C c frac 2 Delta abc nbsp de D displaystyle Delta nbsp ce plosha trikutnika abo ekvivalentno a sin A b sin B c sin C 2 R displaystyle frac a sin A frac b sin B frac c sin C 2R nbsp de R displaystyle R nbsp ce radius kola sho opisuye trikutnik nbsp Figura Lissazhu figura utvorena na osnovi trigonometrichnoyi funkciyi Ce mozhna dovesti rozdilivshi trikutnik na dva pryamokutnih trikutniki i vikoristovuyuchi viznachennya sinusa Teorema sinusiv korisna dlya rozrahunku dovzhin nevidomih storin trikutnika pri vidomih dvoh kutah i dovzhini odniyeyi z jogo storin Cya situaciya ye tipovoyu dlya zadachi triangulyaciyi tehniki viznachennya nevidomih vidstanej shlyahom vimiryuvannya dvoh kutiv iz dvoh tochok na dostupnij vidomij vidstani Teorema kosinusiv Redaguvati Teorema kosinusiv ye uzagalnennyam teoremi Pifagora c 2 a 2 b 2 2 a b cos C displaystyle c 2 a 2 b 2 2ab cos C nbsp abo ekvivalentno cos C a 2 b 2 c 2 2 a b displaystyle cos C frac a 2 b 2 c 2 2ab nbsp V cij formuli kut C displaystyle C nbsp ye protilezhnim do storoni c displaystyle c nbsp Cyu teoremu mozhna dovesti rozdilivshi trikutnik na dva pryamokutnih trikutniki ta zastosuvavshi teoremu Pifagora Teoremu kosinusiv mozhna zastosuvati dlya viznachennya storoni trikutnika yaksho vidomi dovzhini dvoh storin i kut mizh nimi Takozh yiyi mozhna zastosuvati dlya viznachennya kosinusa kuta i vidpovidno znachennya samogo kuta yaksho vidomi dovzhini vsih storin trikutnika Teorema tangensiv Redaguvati Dokladnishe Teorema tangensivVsi nastupni virazi formulyuyut teoremu tangensiv 1 tg A B 2 tg A B 2 a b a b tg A C 2 tg A C 2 a c a c tg B C 2 tg B C 2 b c b c displaystyle frac operatorname tg dfrac A B 2 operatorname tg dfrac A B 2 frac a b a b qquad frac operatorname tg dfrac A C 2 operatorname tg dfrac A C 2 frac a c a c qquad frac operatorname tg dfrac B C 2 operatorname tg dfrac B C 2 frac b c b c nbsp Poyasnennya cih formul na slovah bulo b gromizdkim ale zakonomirnosti sum i riznic dlya dovzhin storin i vidpovidnih protilezhnih kutiv vidno iz teoremi Teorema kotangensiv Redaguvati Dokladnishe Teorema kotangensivYaksho z 1 s s a s b s c displaystyle zeta sqrt frac 1 s s a s b s c nbsp radius vpisanogo kola v trikutnik i s a b c 2 displaystyle s frac a b c 2 nbsp napivperimetr trikutnika todi vsi nastupni formuli opisuyut teoremu kotangensiv 1 ctg A 2 s a z ctg B 2 s b z ctg C 2 s c z displaystyle operatorname ctg frac A 2 frac s a zeta qquad operatorname ctg frac B 2 frac s b zeta qquad operatorname ctg frac C 2 frac s c zeta nbsp Zvidsi viplivaye sho ctg A 2 s a ctg B 2 s b ctg C 2 s c displaystyle frac operatorname ctg dfrac A 2 s a frac operatorname ctg dfrac B 2 s b frac operatorname ctg dfrac C 2 s c nbsp Na slovah teorema polyagaye v tomu sho kotangens polovinnogo kuta dorivnyuye vidnoshennyu napivperimetra vid yakogo vidnyato storonu protilezhnu zadanomu kutu do radiusa vpisanogo kola Periodichni funkciyi Redaguvati nbsp Animaciya aditivnogo sintezu en meandru iz zbilshennyam kilkosti garmonik nbsp Sinusoyidalni bazisni funkciyi znizu mozhut sformuvati pilopodibnu hvilyu zverhu yaksho yih dodati mizh soboyu Vsi bazovi funkciyi matimut vuzli sho zbigayutsya z vuzlami pilopodibnoyi hvili i vsi krim osnovnoyi k 1 displaystyle k 1 nbsp matimut dodatkovi vuzli Kolivannya yaki vidbuvayutsya bilya krayu zubcya pri velikih znachennyah k nazivayutsya yavishem Gibbsa en Trigonometrichni funkciyi takozh mayut vazhlive zastosuvannya u fizici Funkciyi sinusa i kosinusa napriklad vikoristovuyut dlya opisannya garmonichnih kolivan yaki modelyuyut bagato prirodnih yavish taki yak ruh masi zakriplenoyi na pruzhini i dlya malih kutiv ruh mayatnika dlya masi sho visit na nitci Funkciyi sinusa i kosinusa ye odnovimirnimi proyekciyami rivnomirnogo krugovogo ruhu Trigonometrichni funkciyi takozh doveli svoyu korist pri vivchenni zagalnih periodichnih funkcij Harakterna hvilova struktura periodichnih funkcij korisna dlya modelyuvannya yavish takih yak zvukovi abo svitlovi hvili 2 V zagalnih umovah periodichnu funkciyu f x displaystyle f x nbsp mozhna viraziti u viglyadi sumi sinusnih abo kosinusnih hvil za dopomogoyu Ryadu Fur ye 3 Poznachivshi sinusni abo kosinusni bazisni funkciyi yak f k displaystyle varphi k nbsp rozkladannya periodichnoyi funkciyi f x displaystyle f x nbsp bude mati nastupnu formu f t k 1 c k f k t displaystyle f t sum k 1 infty c k varphi k t nbsp Napriklad kvadratnu hvilyu meandr mozhna zapisati u viglyadi ryadu Fur ye f square t 4 p k 1 sin 2 k 1 t 2 k 1 displaystyle f text square t frac 4 pi sum k 1 infty sin big 2k 1 t big over 2k 1 nbsp V animaciyi kvadratnoyi hvili pravoruch mozhna pobachiti sho lishe dekilka termiv vzhe dosit abi stvoriti dobru aproksimaciyu kvadratnoyi formi hvili Superpoziciyu dekilkoh termiv v rozkladanni pilopodibnoyi hvili mozhna pobachiti znizu pid tim malyunkom Div takozh RedaguvatiOberneni trigonometrichni funkciyi Ekzotichni trigonometrichni funkciyi Spisok trigonometrichnih totozhnostej Tablicya integraliv trigonometrichnih funkcij Integralni trigonometrichni funkciyi Trigonometriya Koordinatnij transportirPrimitki Redaguvati a b The Universal Encyclopaedia of Mathematics Pan Reference Books 1976 page 529 530 English version George Allen and Unwin 1964 Translated from the German version Meyers Rechenduden 1960 Farlow Stanley J 1993 Partial differential equations for scientists and engineers vid Reprint of Wiley 1982 Courier Dover Publications s 82 ISBN 978 0 486 67620 3 Arhiv originalu za 20 bereznya 2015 Div priklad Folland Gerald B 2009 Convergence and completeness Fourier Analysis and its Applications vid Reprint of Wadsworth amp Brooks Cole 1992 American Mathematical Society s 77ff ISBN 978 0 8218 4790 9 Arhiv originalu za 12 chervnya 2019 Procitovano 23 lyutogo 2019 Dzherela RedaguvatiKorn G Korn T Spravochnik po matematike dlya nauchnyh rabotnikov i inzhenerov M Nauka 1973 832 s Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1100 s ukr Posilannya RedaguvatiTangens Universalnij slovnik enciklopediya 4 te vid K Teka 2006 Trigonometrichni funkciyi Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 180 594 s FIZMA neT Matematika onlajn Arhivovano 1 kvitnya 2022 u Wayback Machine Trigonometrichni formuli OnlineMSchool Arhiv originalu za 3 bereznya 2021 Procitovano 13 lyutogo 2021 Otrimano z https uk wikipedia org w index php title Trigonometrichni funkciyi amp oldid 40359416
