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 Viznachennya kutiv za dopomogoyu pryamokutnogo trikutnika 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 cos alpha frac AC AB frac b c cos beta frac BC AB frac a c 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 sin alpha frac BC AB frac a c sin beta frac AC AB frac b c 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 mbox tg alpha frac BC AC frac a b mbox tg beta frac AC BC frac b a 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 mbox ctg alpha frac AC BC frac b a mbox ctg beta frac BC AC frac a b Analogichnim chinom mozhna viznachiti trigonometrichni funkciyi na koli z odinichnim radiusom Odin period funkcij sin x displaystyle sin x ta cos x displaystyle cos x sin x sin x ta cos x cos x ce periodichni funkciyi iz periodom 2 p 2 pi tg x operatorname tg x ta ctg x operatorname ctg x mayut period p pi Spivvidnoshennya navedeni nizhche dozvolyayut viraziti znachennya trigonometrichnih funkcij vid dovilnogo dijsnogo argumentu cherez znachennya funkcij dlya argumentu z intervalu 0 p 2 0 pi over 2 sin x cos p 2 x sin x cos left pi over 2 x right cos x sin p 2 x cos x sin left pi over 2 x right tg x ctg p 2 x operatorname tg x operatorname ctg left pi over 2 x right ctg x tg p 2 x operatorname ctg x operatorname tg left pi over 2 x right Osnovni spivvidnoshennya RedaguvatiDokladnishe Spisok trigonometrichnih totozhnostej Nastupne spivvidnoshennya viplivaye iz teoremi Pifagora sin 2 x cos 2 x 1 sin 2 x cos 2 x 1 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 sin left alpha beta right sin alpha cos beta cos alpha sin beta otrimuyemo sin a b sin a cos b cos a sin b sin left alpha pm beta right sin alpha cos beta pm cos alpha sin beta cos a b cos a cos b sin a sin b cos left alpha pm beta right cos alpha cos beta mp sin alpha sin beta tg a b tg a tg b 1 tg a tg b ctg a b ctg a ctg b 1 ctg b ctg a 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 Formuli dlya funkcij podvijnih kutiv Redaguvati sin 2 a 2 sin a cos a displaystyle sin 2 alpha 2 sin alpha cos alpha 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 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 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 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 Formuli dlya funkcij polovinnih kutiv Redaguvati sin a 2 1 cos a 2 cos a 2 1 cos a 2 sin alpha over 2 sqrt 1 cos alpha over 2 cos alpha over 2 sqrt 1 cos alpha over 2 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 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 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 sin A sin B 2 sin A B 2 cos A B 2 sin A pm sin B 2 sin A pm B over 2 cos A mp B over 2 cos A cos B 2 cos A B 2 cos A B 2 cos A cos B 2 cos A B over 2 cos A B over 2 cos A cos B 2 sin A B 2 sin A B 2 cos A cos B 2 sin A B over 2 sin A B over 2 tg A tg B sin A B cos A cos B ctg A ctg B sin B A sin A sin B 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 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 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 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 cos n A cos n A n 2 cos n 2 A sin 2 A n 4 cos n 4 A sin 4 A 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 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 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 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 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 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 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 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 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 cos x 1 x 2 2 x 4 4 x 6 6 n 0 1 n x 2 n 2 n 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 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 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 cdots qquad text pri x lt frac pi 2 end aligned de U n displaystyle U n n te peretvorennya Bustrofedona B n displaystyle B n chisla Bernulli ta E n E n 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 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 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 Zv yazok z eksponentoyu ta kompleksnimi chislami Redaguvati Vikoristovuyuchi vishenavedeni rozkladi v ryadi Tejlora mozhna pokazati sho funkciyi sin displaystyle sin ta cos displaystyle cos ye uyavnoyu ta dijsnoyu chastinami eksponenti chisto uyavnogo chisla e i 8 cos 8 i sin 8 e i theta cos theta i sin theta 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 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 de i 2 1 displaystyle i 2 1 a sh x displaystyle operatorname sh x ta ch x displaystyle operatorname ch x vidpovidno giperbolichni sinus ta kosinus Dlya dijsnogo x displaystyle x 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 Kompleksnij sinus Kompleksnij kosinus Kompleksnij tangensDiferenciyuvannya ta integruvannya Redaguvati f x f x d d x f x frac d dx f x f x d x int f x dx sin x sin x cos x cos x cos x C cos x C cos x cos x sin x sin x sin x C sin x C tg x displaystyle operatorname tg x sec 2 x sec 2 x ln cos x C ln left cos x right C ctg x displaystyle operatorname ctg x cosec 2 x displaystyle operatorname cosec 2 x ln sin x C ln left sin x right C sec x sec x sec x tg x displaystyle sec x operatorname tg x ln sec x tg x C displaystyle ln left sec x operatorname tg x right C cosec x displaystyle operatorname cosec x cosec x ctg x displaystyle operatorname cosec x operatorname ctg x ln cosec x ctg x C displaystyle ln left operatorname cosec x operatorname ctg x right C Zv yazok z diferencialnim rivnyannyam RedaguvatiFunkciyi sin x sin x ta cos x cos x ye rozv yazkami diferencialnogo rivnyannya garmonichnih kolivan d 2 y d x 2 y 0 d 2 y over d x 2 y 0 Vlastivosti ta zastosuvannya RedaguvatiTeorema sinusiv Redaguvati Teorema sinusiv stverdzhuye sho dlya dovilnogo trikutnika zi storonami a displaystyle a b displaystyle b i c displaystyle c ta kutami sho protilezhni tim storonam A displaystyle A B displaystyle B i C displaystyle C 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 de D displaystyle Delta ce plosha trikutnika abo ekvivalentno a sin A b sin B c sin C 2 R frac a sin A frac b sin B frac c sin C 2R de R displaystyle R ce radius kola sho opisuye trikutnik 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 abo ekvivalentno cos C a 2 b 2 c 2 2 a b displaystyle cos C frac a 2 b 2 c 2 2ab V cij formuli kut C displaystyle C ye protilezhnim do storoni c displaystyle c 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 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 radius vpisanogo kola v trikutnik i s a b c 2 displaystyle s frac a b c 2 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 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 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 Animaciya aditivnogo sintezu en meandru iz zbilshennyam kilkosti garmonik 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 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 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 rozkladannya periodichnoyi funkciyi f x displaystyle f x 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 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 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 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 39837725
