5.1 Properties of Inverse Trigonometric Functions.

Transcription

1 Inverse Trignmetricl Functins The inverse f functin f( ) f ( ) f : A B eists if f is ne-ne nt ie, ijectin nd is given Cnsider the e functin with dmin R nd rnge [, ] Clerl this functin is nt ijectin nd s it is nt invertile If we restrict the dmin f it in such w tht it ecmes ne ne, then it wuld ecme invertile If we cnsider e s functin with dmin, nd c-dmin [, ], then it is ijectin nd therefre, invertile The inverse f e functin is defined s, where, nd [,] Prperties f Inverse Trignmetric Functins () Mening f inverse functin (i) (iv) (ii) (iii) ct ct (v) sec sec (vi) ec ec () Dmin nd rnge f inverse functins (i) If, then, under certin cnditin ; ut Agin, nd Keeping in mind numericll smllest ngles r rel numers These restrictins n the vlues f nd prvide us with the dmin nd rnge fr the functin ie, Dmin : [,] Rnge:, (ii) Let, then, under certin cnditins 0 0 {s is decreg functin in [ 0, ]; hence 0 These restrictins n the vlues f nd prvide us the dmin nd rnge fr the functin (, /) (, /) O Y Y O (, /) X (, 0) X

2 Inverse Trignmetricl Functins ie Dmin: [,] Rnge : [ 0, ] (iii) If, then, under certin cnditins Here, R R, Thus, Dmin R ; O Rnge, / (iv) If ct, then ct under certin cnditins, ct R R; ct 0 These cnditins n nd mke the functin, nd nt s tht the inverse functin eists ie, meningful ct ne-ne ct is O (0, /) ct Dmin : R Rnge : ( 0, ) (v) If sec, then sec, where nd 0, (,) Here, Dmin: R (,) Rnge: [0, ] O (,0 ) sec / (vi) If ec, then ec Where nd, 0 (, /) Here, Dmin R (,) Rnge, {0} (, ) O ec

3 Inverse Trignmetricl Functins Functin Dmin (D) Rnge (R) r [,] r, r [,] 0 r [ 0, ] ct ie, R r r, (, ) ie, R r 0 r ( 0, ) (, ) sec, r (, ] [, ) ec, () ( ), Prvided tht 0 r (, ] [, ), 0 r 0,, 0, r, 0 0,, ( ), Prvided tht ( ), Prvided tht, ct (ct), Prvided tht 0 sec (sec ), Prvided tht 0 r ec (ec ), Prvided tht 0 r 0 () ( ), Prvided tht, ( ), Prvided tht ( ), Prvided tht ct(ct ), sec(sec ), Prvided tht r ec (ec ), Prvided tht r () ( ) ( ), ( ) ct () sec ( ) ct sec ( ) sec ec ( ) ec, fr ll [,] - ec, fr ll (, ] [, ) Imprt Tips Prvided ct, fr ll R tht Here;, ec, elng t I nd IV Qudrnt / Here;, sec, ct elng t I nd II Qudrnt I Qudrnt is cmmn t ll the inverse functins III Qudrnt is nt used in inverse functin I IV II I 0

4 Inverse Trignmetricl Functins IV Qudrnt is used in the clckwise directin ie, 0 (7) Principl vlues fr inverse circulr functins Thus, nt etc Nte : Principl vlues fr 0 Principl vlues fr ct ct 0 sec sec 0 ec ec 0 ; nt,, ; ( ) nt re ls written s rc ; ct ( ) nt, rc nd rc respectivel It shuld e nted tht if nt therwise stted nl principl vlues f inverse circulr functins re t e cnsidered (8) Cnversin prpert : Let, ec ec ct sec ec sec ec ct

5 Nte : ct ec sec, fr ll (,] [, ) sec, fr ll (,] [, ) ct, ct fr 0, fr 0 Inverse Trignmetricl Functins ec (9) Generl vlues f inverse circulr functins: We knw tht if is the smllest ngle whse e is, then ll the ngles whse e is cn e written s n ( ), where n 0,,, Therefre, the generl vlue f dented Thus, we hve n n cn e tken s n () The generl vlue f n π π nπ ( ) α,, if α nd α Similrl, generl vlues f ther inverse circulr functins re given s fllws: n, ; If, 0 n, R ; If, ct n, R ; If ct, 0 sec n, r ; If sec,0 nd ec n ( ) n, r ; If ec, nd 0 is Emple: Slutin: Emple: Slutin: The principl vlue f sec [sec( 0 sec )] 0 [sec( 0 )] sec is [Rrkee 99] (sec 0 ) [MP PET 99] Emple: The principl vlue f is [MP PET 99]

6 Slutin: Inverse Trignmetricl Functins Emple: The principl vlue f is [IIT 98] Slutin: The principl vlue f [( )] Nne f these Emple: Cnsidering nl the principl vlues, if ( ) ct, then is equl t Slutin: Put ct ct Put then Als, Emple: If [ ( 00 )], then ne f the pssile vlue f is Slutin: [( 00 )] [ (0 0 )] [ 0 ] [ ( 80 0 )] 0 Emple: 7 Vlue f is [Rrkee 000] Slutin: Emple: 8 Slutin: 0 The equtin 0 0 hs [AMU 999] N slutin Onl ne slutin Tw slutins Three slutins Given equtin is ( ), which is nt pssile s 0, Emple: 9 If, then [EAMCET 99]

7 Inverse Trignmetricl Functins 7 Slutin: Emple: 0 If, 0 then the smllest intervl in which lies is Slutin: 0 0 We knw 0 Emple: If fr 0, then equls [IIT Screening 00] Slutin: We knw tht, Accrding t questin,, ( 0 ) 0 ( ) 0 0 nd, ut 0 S, Emple: If ct, then is [Krntk 999; Rrkee 999] 0 Slutin: ct ct ; Clerl, Emple: The vlue f (ct ) 987] ) / is / ) ( ( ) / [MP PET 00; UPSEAT ( ( / ) Slutin: (ct ) Emple: The numer f rel slutins f ( ) is [IIT Screening999] Zer One Tw Infinite

8 Inverse Trignmetricl Functins 8 Slutin: ( ) ( ) is defined, when ( ) 0 (i) is defined, when 0 ( ) r 0 ( ) 0 (ii) Frm (i) nd (ii), ( ) 0 r 0 nd Hence, numer f slutins is Frmule fr Sum nd Difference f Inverse Trignmetric Functin () ; If 0, 0 nd () ; If 0, 0 nd () ; If 0, 0 nd () ; If () ; If 0, 0 nd () ; If 0, 0 nd z z (7) z z z S S S (8) n, S S S where (9) ct (0) S k dentes the sum f the prducts f ct ct ct ct ct () { } ;,, If, nd r if 0 nd, n tken k t time () { }, If 0, nd () { }, If ; 0 nd

9 Inverse Trignmetricl Functins 9 () { }, If ; nd if r 0 nd () { }, If 0, 0 nd () { }, If 0, 0 nd (7) { }, If, nd 0 (8) { }, If, nd 0 (9) { }, (0) { }, If,, nd If 0, 0 nd If z, then z z If z, then z z If z, then z z Imprt Tips If z, then z z z If z, then z z If z, then z z If z, then z z If, then If, then If, then If ct ct, then If, then Emple: The vlue f is [AMU 00] 7 7

10 Slutin: Emple: 00] Inverse Trignmetricl Functins 0 [MP PET 997, 00; UPSEAT 00; Krntk CET Slutin: Emple: 7 If, then is equl t [Rrkee 99] Slutin: 0 Therefre, 9 9 Emple: 8 ct is equl t 99; Krntk CET 99] 9 9 Slutin: ct ct ct ct 9 () ct 9 () ct ct () [MP PET Emple: 9 If C, then C [P CET 999] Slutin: Given, C C 9 C 9 Emple: 0 If f ( ), then f f Sltuin: (,d) f ( ) f f

11 Inverse Trignmetricl Functins Emple: Slutin: Emple: ( ), ccrding s r 0 if, which hlds fr if, which hlds fr 8 ( ) 0 If nd, then Nne f these Slutin: Since 8, ( ) Emple: If, then 9 Slutin: Nne f these 9 ( ) ( )(9 ) 9 ( ) Emple: The numer f slutins f is 0 Infinitie Slutin: ( ) r 8, (nt )

12 Inverse Trignmetricl Functins Emple: If, then [UPSEAT 999; MP PET 99] ct Slutin: We hve Emple: If,, c e psitive rel numers nd the vlue f c ( c), then is ( c) ( c) c c 0 c Nne f these Slutin: ( c) ( c) c( c) c c Let s c c s s c s ( s) ( s) ( cs) s s cs cs s cs cs ( c) cs s 0 [ cs ( c)] ( c c) s Trick : Since it is n identit s it will e true fr n vlue f,,c Let c then, 0 Emple: 7 All pssile vlues f p nd q fr which p, q p p q hlds, is q, p 0 p, q Nne f these Slutin: p p q p p q p p p p q q 0 q q q Inverse Trignmetric Rtis f Multiple Angles () ( ), If () ( ), If

13 () ( ), If Inverse Trignmetricl Functins () ( ), If () ( ), If () ( ), If (7) ( ), If 0 (8) ( ), if 0 (9) ( ) () ( ), If If (0) ( ), If (), if (), If (), If (), If (), If (7), If (8) (9) () () (), If, If ( ), If 0, If 0 (0), If () () ()

14 Inverse Trignmetricl Functins Emple: 8 ( ) (ec ), then [UPSEAT 00] Slutin: ( ) ( ec ) Emple: 9 The slutin set f the equtin is [AMU 00] Sltuin: {, } {, } {,, 0} {,,0} 0 ( )( ) 0 {,,0 } Emple: 0 is equl t [Kurukshetr CEE 00] 0 Slutin: Emple: If, then Slutin: Put, nd, then reduced frm is [EAMCET 989] ( ) ( ) ( ) Tking n th sides, we get ( ) Sustituting these vlues, we get Emple: Slutin: Emple: () () Nne f these [IIT 98] [Rrkee 98]

15 Inverse Trignmetricl Functins Slutin: Emple: The vlue f ( ) () [AMU 999] Slutin: [ ( )] [ ( )] 9 Emple: Slutin: [ ] Let equl t [MP PET 999] t t t t, where t ( t ) t Emple: is equl t 9 Slutin: Since, S, (0 9 ) 9 (9 9 ) 0

16 Emple: 7 The frmul hlds nl fr Slutin: If, LHS, Emple: 8 Inverse Trignmetricl Functins R (, ] [, ) RHS S, the fmul des nt hld If, the ngle n the LHS is in the secnd qudrnt while the ngle n the RHS is (ngle in the furth qudrnt), which cnnt e equl If, the ngle n the LHS is in the secnd qudrnt while the ngle n the RHS is (ngle in the first qudrnt) nd these tw m e equl If 0, the ngle n the LHS is psitive nd tht n the RHS is negtive nd the tw cnnt e equl is independent f, then [, ) [,] (, ] Nne f these Slutin: Let Then ( ) ( ) If, independent f If, [( )] independent f, ut, nd frm the principl vlue f, Hence,,, Als t, The given functin cnst if, ie, [, ) Emple: 9 The numer f psitive integrl slutins f the equtin r 0 ct is One Tw Zer Nne f these Slutin: r r As, re psitive integers,, nd crrespnding, 7 Emple: 0 Slutins re (, ) (, ),(,7), nd re three ngles given ( ), nd Then Nne f these

17 Inverse Trignmetricl Functins 7 Slutin: (,c) 8 8 ) ( 7 Als, S, Agin elngs t the first qudrnt nd is in the secnd qudrnt Emple: c sec sec is equl t ) )( ( ) )( ( ) )( ( Nne f these Sltuin: Let,, c sec sec + ) ( ] [ ) ( ] [ (rtinlized) }] { } { [ ) )( ( )] ( [ ***