Mathematics. Trigonometrical Ratio, Functions & Identities
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1 Mthemtics Tigmeticl Rti, Fuctis & Idetities
2 Tble f tet Defiitis stems f Mesuemet f gles Relti betwee Thee stems f Mesuemet f gle Relti betwee c d gle 5 Tigmeticl Rtis Fuctis 6 Tigmeticl Rtis f llied gles 7 Tigmeticl Rtis f Vius gles 8 Tigmeticl Rtis i tems f ech the 9 Tigmeticl Rtis f um d Diffeece f Tw gles 0 Tigmeticl Rtis f um d Diffeece f Thee gles Tsfm the Pduct it um Diffeece Tigmeticl Rtis f multiple f gle Tigmeticl Rtis f sub multiple f gle Mimum d Miimum vlue f cs + b si 5 ditil Tigmeticl Idetities
3 Defiiti ( gle: The mti f evlvig lie i ple fm its iitil psiti (iitil side t the fil psiti (temil side is clled gle The ed pit O but which the lie ttes is clled the vete f the gle Temil ( Mesue f gle: The mesue f gle is the mut f tti fm the iitil side t the temil side ( ese f gle: The sese f gle is detemied b the diecti f tti f the iitil side it the temil side The sese f gle is sid t be psitive egtive ccdig s the iitil side ttes i ticlckwise clckwise diecti t get the temil side Psitive Negtive ( Right gle: If the evlvig sttig fm its iitil psiti t fil psiti descibes e qute f cicle The we s tht the mesue f the gle fmed is ight gle (5 Qudts: Let X' OX d YOY ' be tw lies t ight gles i the ple f the ppe These lies divide the ple f ppe it fu equl pts Which e kw s qudts The lies X' OX d YOY ' e kw s -is d -is These tw lies tke tgethe e kw s the c-dite es Y II qudt I qudt (6 gle i stdd psiti: gle is sid t be i stdd psiti if its vete ccides with the igi O d the iitil side ccides with OX ie, the psitive diecti f -is III qudt O IV qudt X (7 gle i qudt: gle is sid t be i pticul qudt if the temil side f the gle i stdd psiti lies i tht qudt (8 Qudt gle: gle is sid t be qudt gle if the temil side ccides with e f the es
4 stem f Mesuemet f gles Thee e thee sstem f mesuig gles ( egesiml Eglish sstem: Hee ight gle is divided it 90 equl pts kw s degees Ech degee is divided it 60 equl pts clled miutes d ech miute is futhe divided it 60 equl pts clled secds Theefe, ight gle = 90 degee ( miutes ( 60 ' 60 secd ( 60' ' ' ( etesiml Fech sstem : It is ls kw s Fech sstem, hee ight gle is divided it 00 equl pts clled gdes d ech gde is divided it 00 equl pts, clled miutes d ech miute is futhe divided it 00 secds Theefe, ight gle = 00 gdes ( 00 gde = 00 miutes ( 00' miute = 00 secds ( 00' ' g ( icul sstem: I this sstem the uit f mesuemet is di Oe di, witte s c, is the mesue f gle subteded t the cete f cicle b c f legth equl t the dius f the cicle P side cicle f dius hvig cete t O Let be pit the cicle Nw cut ff c P whse legth is equl t the dius f the cicle The b the defiiti the mesue f di ( c OP is
5 Relti betwee Thee stems f Mesuemet f gle Let D be the umbe f degees, R be the umbe f dis d G be the umbe f gdes i gle Nw, 90 = ight gle D D ight gles ight gle D ight gles (i 90 gi, dis = ight gles di R dis ight gles R R ight gles ight gles (ii d 00 gdes = ight gle gde G gdes G ight gles 00 ight gle 00 G ight gles (iii 00 Fm (i, (ii d (iii we get, D G R π This is the equied elti betwee the thee sstems f mesuemet f gle Nte: Oe di 80 dis 80 di =
6 Relti betwee c d gle If s is the legth f c f cicle f dius, the the gle (i dis subteded b this c t the s cete f the cicle is give b s θ ie, c = dius gle i dis ectil e: Let O be sect hvig cetl gle the sect O is give b Imptt Tips θ d dius The e f The gle betwee tw csecutive digits i clck is 0 (= /6 dis The hu hd ttes thugh gle f 0 i e hu The miute hd tte thugh gle f 6 i e miute s 5 Tigmeticl Rtis Fuctis I the ight gled tigle OMP, we hve bse = OM =, pepedicul =PM = d hpteuse = OP = We defie the fllwig tigmetic ti which e ls kw s tigmetic fucti Pepedicu l si Hpteues t sec Pepedicu l se Hpteues se se cs Hpteues se ct, Pepedicu l csec Hpteues Pepedicu l O Y M P(, X 5
7 ( Relti betwee tigmetic tis (fucti (i si c sec (ii t ct (iii cs sec (iv t si cs (v ct cs si ( Fudmetl tigmetic idetities (i si cs (ii t sec (iii ct csec Imptt Tips If sec + t, the sec t If cesc ct, the csec ct ( ig f tigmeticl tis fuctis: Thei sigs depeds the qudt i which the temil side f the gle lies (i I fist qudt: 0, 0 si 0,cs 0,t 0, csec 0, sec 0 d ct 0 Thus, i the fist qudt ll tigmetic fuctis e psitive 6
8 (ii I secd qudt: sec 0 d ct 0 ll thes e egtive 0, 0 si 0,cs 0,t 0,csec 0, Thus, i the secd qudt si d csec fucti e psitive d (iii I thid qudt: 0, 0 si 0, cs 0, t 0, csec 0, sec 0 d ct 0 ecept tget d ctget Thus, i the thid qudt ll tigmetic fuctis e egtive (iv I futh qudt: 0, 0 si 0, cs 0, t 0, csec 0, sec 0 d ct 0 Thus, i the futh qudt ll tigmetic fuctis e egtive ecept cs d sec I bief : cude id t memise the sigs f tigmeticl ti i diffeet qudt "dd ug T ffee" Y II I qudt < 0, > 0 > 0, > 0 si d csec e ll e psitive psitive III qudt O IV qudt T < 0, < 0 > 0, < 0 X X t d ct cs d sec Y Imptt Tips Fist detemie the sig f the tigmetic fucti If is mesued fm X OX ie, {(, } the eti the igil me f the fucti If is mesued fm Y OY ie,,, the chge sie t csie, csie t sie, tget t ctget, ct t t, sec t csec d csec t sec 7
9 ( Vitis i vlues f tigmetic fuctis i diffeet qudts: Let X' OX d YOY ' be the cdite es Dw cicle with cete t igi O d dius uit Let M (, be pit the cicle such tht OM the cs d si ; cs d si f ll vlues f II-Qudt ( I-Qudt ( si deceses fm t 0 si iceses fm 0 t cs deceses fm 0 t cs deceses fm t 0 t iceses fm t 0 t iceses fm 0 t ct deceses fm 0 t ct deceses fm t 0 sec iceses fm t sec iceses fm t csec iceses fm t csec deceses fm t III-Qudt (T si deceses fm 0 t cs iceses fm t 0 t iceses fm 0 t ct deceses fm t 0 sec deceses fm t IV-Qudt ( si iceses fm t 0 cs iceses fm 0 t t iceses fm t 0 ct deceses fm 0 t sec deceses fm t csec iceses fm t csec deceses fm t (, Y (0, M (, X N (0, X Nte: d e tw smbls These e t el umbe Whe we s tht t iceses fm 0 t f s vies fm 0 t it mes tht t iceses i the itevl 0, d it ttis lge psitive vlues s teds t imill f the tigmetic fuctis 8
10 6 Tigmeticl Rtis f llied gles Tw gles e sid t be llied whe thei sum diffeece is eithe ze multiple f ( Tigmetic tis f ( : Let evlvig sttig fm its iitil psiti OX, tce ut gle XO Let P(, be pit O such tht OP = Dw PM fm P -is gle clckwise sese Let P ' be pit XO ' i the O ' such tht OP' OP lel M d M cicide d OMP is cguet t OMP ' the P ' e (, si( si ; cs( cs ; t( t Tkig the ecipcl f these tigmetic tis; csec( csec, sec( sec d ct( ct O Y 90 P(, M P (, X Nte: fucti f ( is sid t be eve fucti if f( f( f ll i its dmi fucti f ( is sid t be dd fucti if f( f( f ll i its dmi si, t, ct, csec e dd fuctis d cs, sec e eve fuctis ( Tigmetic fucti f (90 : Let the evlvig lie, sttig fm O, tce ut cute gle OP, equl t Fm pit P, dw PM t O Thee gles f tigle e tgethe equl t tw ight gles, d sice OMP is ight gle, the sum f the tw gles MOP d OPM is ight gle OPM 90 [Whe the gle OPM is cside, the lie PM is the bse d MO is the pepedicul ] MO PM si( 90 si MPO cs OP cs, cs( 90 cs MPO si OP si PO PO MO PM t( 90 t MPO ct OP ct, ct( 90 ct MPO t OP t PM MO PO csec(90 csec MPO sec OP sec, MO PO sec( 90 sec MPO csec OP csec PM O M P 9
11 ( Tigmetic fucti f (90+ : Let evlvig O sttig fm its iitil psiti OX, tce ut gle XO d let the evlvig O sttig fm the sme iitil psiti OX, fist tce ut gle s t cicide with O d the it evlves thugh gle f ticlckwise diecti t fm gle XO ' 90 Let P d OP OP' P ' be pits O d Dw pepedicul PM d PM ' fm P d OX Let the cdites f P be (, The clel, OM' PM d P' M' OM O ' espectivel such tht 90 i P ' espectivel OM d PM the cdites f P ' e, M' P' OM ' si( 90 cs, cs( 90 si OP' OP' M' P' t( 90 ct, ct( 90 t, sec( 90 csec, csec(90 sec OM' (, O Y Y M P(, X llied gles ( θ (90 ( 90 θ (80 θ (80 θ (70 θ (70 θ ( 60 θ Tig Rti π θ π θ ( π θ ( π θ π θ π θ ( π θ si si cs cs si si cs cs si cs cs si si cs cs si si cs t t ct ct t t ct ct t Imptt Tips si 0, cs ( si( ( si, cs( ( cs si ( / = ( si, if is eve cs ( ( / cs, if is dd si, if isdd cs, if iseve 0
12 7 Tigmeticl Rtis f Vius gles 0 /6 / / / / si 0 / / / 0 0 cs / / / 0 0 t 0 / Tigmeticl Rtis i tems f Ech the si si cs t ct sec csec si t cs t ct sec sec csec cs si cs t ct ct sec csec csec t si si cs cs t ct sec csec ct si si cs cs t ct sec csec sec si cs t ct ct sec csec csec csec si cs t t ct sec sec csec
13 Imptt Tips Vlues f sme stdd gles si5 cs 75 ; cs5 si75 ; t 5 ct 75 ; si8 cs 7 5 ; cs 6 si5 5 ; t 75 ct5 si cs 67, cs si67 ; ct t 67 t ct 67 9 Fmule f the Tigmetic Rtis f um d Diffeeces f Tw gles ( si( si cs cs si ( si( si cs cs si ( cs( cs cs si si ( cs( cs cs si si (5 t t t( t t (6 t t t( t t (7 ct( ct ct ct ct (8 ct( ct ct ct ct
14 (9 cs cs si si si( si( (0 si cs si cs cs( cs( ( cs cs si( cs cs si cs cs si cs si cs si t t m, ( si si si( ct ct, m 0 Fmule f the Tigmetic Rtis f um d Diffeeces f Thee gles ( si si si si cs cs cs si cs cs cs si si( si t t t t t (t cs cs cs ( ( si si cs si cs si cs si si cs cs cs cs( t t t t t t ( cs cs cs cs( ( t t t t t t t t t t t t t( ( ct ct ct ct ct ct ct ct ct ct ct ct ct( I geel; (5 si( = ( cs cs cs 7 5 (6 ( cs cs cs cs( 6 (7 t( 6 7 5
15 Whee; t t t = The sum f the tgets f the septe gles t t t t = The sum f the tgets tke tw t time t t t t t t = um f tgets thee t time, d s If, the t, t, t, 5 5 (8 si cs ( t t t (9 cs cs ( t t (0 t t t 5 t t t t ( si cs cs ( t t t t 5 t 6 t 5 6 ( si cs cs ( t t t t 5 t 6 t 5 6 ( si( si( si( si( ( = si{ ( ( / } si( / si( / ( cs( cs( cs( cs( ( = cs ( si si Fmule t Tsfm the Pduct it um Diffeece ( si cs si( si( ( cs si si( si( ( cs cs cs( cs(
16 ( si si cs( cs( Let d D D D The, d Theefe, we fid ut the fmule t tsfm the sum diffeece it pduct (5 (6 (7 (8 D D si si D si cs D D si si D cs si D D cs cs D cs cs D D cs cs D si si D D si si Imptt Tips =, if si( 60 si si( 60 si t( 60 t t(60 t cs( 60 si cs cs cs cs cs,if si =, if ( cs cs(60 cs Tigmetic Rti f Multiple f gle ( si si cs t t ( cs cs si ( t t t cs si t t ; whee ( 5
17 ( si si si si(60 si si(60 (5 cs cs cs cs(60 cs cs(60 t t (6 t t(60 t t(60, whee / 6 t (7 si si cs cs si (8 cs 8 cs 8 cs (9 t t t 5 (0 si 5 6 si 0 si 5 si 6 t t 5 ( cs 5 6 cs 0 cs 5 cs Tigmetic Rti f ub-multiple f gle ( si cs si si cs si ie,, If / /, thewise ( si cs si (si cs si ie, 5, If / /, thewise ( (i t t cs cs, whee ( t cs si (ii ct cs cs, whee cs si 6
18 The mbiguities f sigs e emved b lctig the qudts i which lies u c fllw the fllwig figue, si + cs is +ve si is+ve si si + cs cs is ve is +ve si + cs is +ve si cs is ve si + cs is ve si cs is ve ( t cs ; whee ( cs (5 ct cs ; whee cs Imptt Tips fmul tht gives the vlue f fmul tht gives the vlue f fmul tht gives the vlue f si cs i tems f si shll ls give the vlue f sie f i tems f cs shll ls give the vlue f cs f t i tems f t shll ls give the vlue f t f ( 7
19 Mimum d Miimum Vlue f cs + b si Let cs (i d b si (ii quig d ddig (i d (ii, the b, b si b cs = (si cs cs si = si( ut si, si( ; The si( Hece, b si b cs b The the getest d lest vlues f si b cs e espectivel b d b Nte: si csec, f eve el cs t sec, f eve el ct, f eve el Imptt Tips Use f (igm d (Pie tti si( si cs cs si, cs( cs cs si si, t t t( ( detes summti t t si si( si( tems ( detes pduct si si si[ ( / ] si[ / ] si( si( / si cs cs( cs( tems cs si cs( si si / cs / si / cs / cs[ ( / ] si[ / ] si[ / ] cs cs cs cs( cs cs cs si si si si( si si si t t t 8 ct 8 ct 8
20 5 ditil Tigmeticl Idetities We hve ceti tigmetic idetities Like, si cs d t sec etc uch idetities e idetities i the sese tht the hld f ll vlue f the gles which stisf the give cditi mg them d the e clled cditil idetities If,, dete the gles f tigle, the the elti + + = ebles us t estblish m imptt idetities ivlvig tigmetic tis f these gles ( If + + =, the + =, + = d + = ( If + + =, the si( si( si imill, si( si( si d si( si( si ( If, the cs( cs( cs imill, cs( cs( cs d cs( cs( cs ( If + + =, the t( t( t imill, t( t( t d t( t( t (5 If, the d d si si cs, cs cs si, t t ct 9
21 ll pblems cditil idetities e bdl divided it the fllwig thee tpes Idetities ivlvig sie d csie f the multiple sub-multiple f the gles ivlved Wkig Methd tep (i: Use D fmule tep (ii: Use the give elti ( + + = i the epessi btied i step-(i such tht fct c be tke cmm fte usig multiple gles fmule i the emiig tem tep (iii: Tke the cmm fct utside tep (iv: gi use the give elti ( + + = withi the bcket i such me s tht we c ppl D fmule tep (v: Fid the esult ccdig t the give ptis Idetities ivlvig sques f sie d csie f multiple sub-multiples f the gles ivlved Wkig Methd tep (i: ge the tems f the idetit such tht eithe si si si( si( cs si cs( cs( c be used tep (ii: Tke the cmm fct utside tep (iii: Use the give elti ( withi the bcket i such me s tht we c ppl D fmule tep (iv: Fid the esult ccdig t the give ptis Idetities f tget d ctget f the gles Wkig Methd tep (i: Epess the sum f the tw gles i tems f thid gle b usig the give elti ( tep (ii: Tkig tget ctget f the gles f bth the sides 0
22 tep (iii: Use sum d diffeece fmule i the left hd side tep (iv: Use css multiplicti i the epessi btied i the step (iii tep (v: ge the tems s pe the esult equied Imptt Tips Methd f cmped d divided If b q p, the b cmped d divided We c wite b b q p q p b b p q p q b b q p q p b b p q p q
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