Chapter two Functions

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1 Chaptr two Functions -- Eponntial Logarithm functions Eponntial functions If a is a positiv numbr is an numbr, w dfin th ponntial function as = a with domain - < < ang > Th proprtis of th ponntial functions ar. If a > a >.. a. a = a +.. a / a = a -.. ( a ) = a.. 5. ( a. b ) = a. b. 6. a a ( a ). 7. a - = / a a = / a a = a =. 9. a =, a =, a - =, whr a >. a =, a - =, whr a <. Th graph of th ponntial function = a is Logarithm function If a is an positiv numbr othr than, thn th logarithm of to th bas a dnotd b = log a whr > At a = =.7888, w gt th natural logarithm dnotd b = ln Lt, > thn th proprtis of logarithm functions ar. = a = log a = = ln.. log = ln.. log a = ln / ln a. ١

2 . ln (.) = ln + ln. 5. ln ( / ) = ln ln. 6. ln n = n. ln. 7. ln = log a a = ln = log a =. 8. a =. ln a. 9. ln =. Th graph of th function = ln is X 5 Application of ponntial logarithm functions W tak Nwton's law of cooling T T S = ( T T S ) t k whr T is th tmpratur of th objct at tim t. T S is th surrounding tmpratur. T is th initial tmpratur of th objct. k is a constant. EX-- Th tmpratur of an ingot of mtal is 8 o C th room tmpratur is o C. Aftr twnt minuts, it was 7 o C. Sol. a) What is th tmpratur will th mtal b aftr minuts? b) What is th tmpratur will th mtal b aftr two hours? c) Whn will th mtal b o C? T T S ( T T S ) tk 5 6 k k ln 5 ln 6.9 a ) T 6 (.9 ) 6 * o C T 65.6 o C b ) T T S 6 (.9 ) 6 *.5. o C T. o C c ) 6.9 t.9 t ln 6 t. hrs. ٢

3 -- Trigonomtric functions Whn an angl of masur θ is placd in stard position at th cntr of a circl of radius r, th trigonomtric functions of θ ar dfind b th quations Sin Sin, Cos, tan r csc r sc Cot Cos o r θ Th following ar som proprtis of ths functions ) ) ) ) 5 ) 6 ) 7 ) 8 ) 9 ) ) Sin Cos tan sc Cot csc Sin( ) Sin.Cos Cos.Sin Cos( ) Cos.Cos Sin.Sin tan tan tan( ) tan.tan Sin Sin.Cos Cos Cos Sin Cos Cos Cos Sin Sin( ) Cos Cos( ) Sin Sin( ) Sin Cos( ) Cos tan( ) tan Sin.Sin [Cos( ) Cos( )] Cos.Cos [Cos( ) Cos( )] Sin.Cos [ Sin( ) Sin( )] ٣

4 ) ) Sin Sin Sin.Cos Sin Sin Cos.Sin Cos Cos Cos.Cos Cos Cos Sin.Sin θ Π / 6 Π / Π / Π / Π Sinθ / / / Cosθ / / / - tanθ / Graphs of th trigonomtric functions ar.5.5 -Л -Л Л Л Sin ٤

5 .5.5 -Л -Л Л Л Cos -π -π π π tan n ٥

6 ٦ -π -π π π n Cot -π -π π π - or n Sc

7 -π -π π π - csc n or Whr n,,,,... EX- - Solv th following quations, for valus of θ from o to 6 o inclusiv. a) tan θ = Sin θ b) + Cos θ = Sin θ Sol.- Sin a ) tan Sin Sin Cos Sin ( Cos ) ithr o o Sin,8,6 or o o Cos 6, Thrfor th rquird valus of θ ar o,6 o,8 o, o,6 o. o b ) Cos.Sin Cos ( Cos ) ( Cos )( Cos ) o ithr Cos 6, o o or Cos 8 Thr th roots of th quation btwn o o. 6 o ar 6 o,8 o ٧

8 EX-- Prov th following idntitis a ) Csc tan.sc Csc.Sc EX-- If tan θ = 7/, find without using tabls th valus of Scθ Sinθ. Sol.- 7 tan r 7 5 r 5 ٧ 7 Sc Sin r 5 Sol.- b ) c ) a ) b ) c ) Cos Sin Cos Sin Sc Csc tan Cot tan Cot Sc Csc Sin L.H.S. Csc tan.sc. Sin Cos Cos Cos Sin. Csc.Sc.H.S. Sin.Cos Sin Cos L.H.S. Cos Sin ( Cos Sin ).( Cos Sin ) Cos Sin.H.S. Sc Csc L.H.S. Cos Sin tan Cot Sin Cos Sin Cos Cos Sin Sin Cos Sin.Cos tan Cot..H.S. Sin Cos Sc Csc Sin.Cos ٢٤ EX-5- Simplif Sol.- a a a whn Csc a a a.csc Cot. a tan. EX-6- Eliminat θ from th quations i) = a Sinθ = b tanθ ii) = Scθ = Cosθ Sol.- ٨

9 i ) a a.sin Sin Csc a b b tan tan Cot b a b Sinc Csc Cot ii ) Sc Cos Cos Cos Sin 8 EX-7- If tan θ tan β =, show that Cos θ Cos β =. Sol. tan tan Sc ( Sc ) Sc Sc Cos Cos Cos Cos Q.E.. EX-8- If a Sinθ = p b Cosθ b Sinθ = q + a Cosθ.Show that a +b = p +q Sol.- p a.sin b.cos p q ( asin bcos ) a q b.sin a.cos ( bsin acos ) ( Sin Cos ) b ( Cos Sin ) a EX-9- If Sin A = / 5 Cos B = /,whr A is obtus B is acut. Find, without tabls, th valus of a) Sin ( A B ), b) tan ( A B ), c) tan ( A + B ). Sol. - b 5 A - B 5 ٩

10 a ) b ) c ) Sin( A B ) SinA.CosB CosA.SinB tan A tan B tan( A B ) tan A.tan B tan A tan B tan( A B ) tan A.tan B EX- Prov th following idntitis a ) b ) c ) d ) Sin( A B ) Sin( A B ).SinA.CosB Sin( A B ) tan A tan B CosA.CosB ScA.ScB.CscA.CscB Sc( A B ) CscA.CscB ScA.ScB Sin Cos Cot Sin Cos ١٠

11 Sol.a ) L.H.S. Sin ( A B ) Sin ( A B ) SinA.CosB CosA.SinB SinA.CosB CosA.SinB b) c) d).sina.cosb.h.s. Sin ( A B ) SinA.CosB CosA.SinB.H.S. CosA.CosB CosA.CosB tan A tan B L.H.S.... ScA.ScB.CscA.CscB.H.S CosA CosB SinA SinB CscA.CscB ScA.ScB.. SinA SinB CosA CosB CosA.CosB SinA.SinB Cos ( A B ) Sc ( A B ) L.H.S. Sin Cos Sin.Cos ( Cos Sin ) L.H.S. Sin Cos Sin.Cos ( Cos Sin ) Sin.Cos Cos Cos Cot.H.S. Sin.Cos Sin Sin EX- Find, without using tabls, th valus of Sin θ Cos θ, whn a) Sinθ = / 5, b) Cos θ = /, c) Sin θ = - /. Sol. a) ٥ 5 θ θ ٤ - Sin.Sin.Cos..( ) Cos Cos Sin ( ) ( ) ١١

12 b) ١٣ 5 θ ١٢ θ -5 ١٣ 5 ).( ) Cos Cos Sin ( ) ( ) 69 Sin.Sin.Cos ( c) -١ θ θ - - ).( ) Cos Cos Sin ( ) ( ) Sin Sin.Cos ( EX-- Solv th following quations for valus of θ from o to 6o inclusiv a) Cos θ + Cos θ + =, b) tan θ. tan θ = Sol.- ١٢

13 a ) b ) Cos Cos Cos Cos Cos(.Cos ) o o ithr Cos 9,7 o o or Cos, o o o o 9,,,7 tan.tan.tan.tan. tan 9 tan o o ithr tan 8.,98. o o or tan 6.6,.6 8. o,6.6 o,98. o,.6 o -- Th invrs trigonomtric functions Th invrs trigonomtric functions aris in problms that rquir finding angls from sid masurmnts in triangls Sin Sin Sin 9 9 ١٣

14 - - Cos 8 π -π tan 9 9 ١٤

15 π π -π Cot - - Sc, ١٥

16 π π - -π -π Csc, Th following ar som proprtis of th invrs trigonomtric functions. Sin ( ) Sin Cos ( ) Cos Sin Cos tan ( ) tan Cot tan Sc Cos Csc Sin Sc ( ) Sc notd that ( Sin ) Csc Sin Sin ١٦

17 EX-- Givn that Sin, find Csc,Cos, Sc,tan,, Cot ٢ Sin Csc ١ Sin,Cos, Sc r,tan,cot EX- Evaluat th following prssions a ) Sc( Cos ) b ) Sin Sin ( ) c ) Cos ( Sin 6 ) Sol.- Sol.- a ) Sc( Cos ) Sc b ) Sin Sin ( ) ( ) c ) Cos ( Sin ) Cos ( ) 6 EX-5- Prov that Sol. a ) b ) a ) Sc Cos b ) Sin ( ) Sin Lt Sc Sc Cos Cos Sc Cos Lt Sin Sin( ) Sin Sin ( ) Sin ( ) Sin ١٧

18 -- Hprbolic functions Hprbolic functions ar usd to dscrib th motions of wavs in lastic solids ; th shaps of lctric powr lins ; tmpratur distributions in mtal fins that cool pips tc. Th hprbolic sin (Sinh) hprbolic cosin (Cosh) ar dfind b th following quations u u Sinhu ( ) u u Sinhu tanh u u u Coshu Schu u u Coshu Cosh u Sinh u tanh u Sch u Coshu Cosh( u ) Coshu Cosh Sinh( Sinhu Coshu Cosh( ) Cosh.Cosh Sinh.Sinh.Cosh u u u ( ) Coshu Cothu Sinhu Cschu Sinhu Coth u Csch u Sinh ) Sinh.Cosh Cosh.Sinh Cosh Cosh Sinh Cosh Cosh Sinh.Sinh u u u Coshu Sinhu Sinh( u ) Sinhu Sinh u u u u Cosh =Sinh =Csch =Csch ١٨

19 =Cosh =Sch =Coth =tanh =Coth - Sinh Cosh tanh Coth Sch Csch or EX-6- Lt tanh u = - 7 / 5, dtrmin th valus of th rmaining fiv hprbolic functions. Sol.- ١٩

20 Cothu tanh Coshu Schu Sinhu tanh u Coshu Cschu u tanh u Sch Sinhu 5 7 u Sch Sinhu 5 u Schu Sinhu 5 7 EX-7- writ th following prssions in trms of ponntials. Writ th final rsult as simpl as ou can a ) Cosh(ln ) b ) tanh(ln ) Sol.- c ) Cosh5 Sinh5 a ) b ) c ) d ) Cosh(ln ) tanh(ln ). ln ln ( Sinh Cosh ) d ) ( Sinh Cosh ) ln ln ln Cosh5 Sinh5 5 ln EX-8- Solv th quation for Cosh = Sinh + /. Sol. - Cosh Sinh ln ln ln EX-9 Vrif th following idntit a) Sinh(u+v)=Sinh u. Cosh v + Cosh u.sinh v b) thn vrif Sinh(u-v)=Sinh u. Cosh v - Cosh u.sinh v Sol.- ٢٠

21 a ) b ).H.S. L.H.S. Sinhu.Coshv Coshu.Sinhv u u v v u u v v. uv ( uv ) Sinh( u v ) L.H.S. Sinh( u ( v )) Sinhu.Cosh( v ) Coshu.Sinh( v ) Sinhu.Coshv Coshu.Sinhv.H.S. EX- Vrif th following idntitis a ) Sinhu.Coshv Sinh( u v ) Sinh( u v ) b ) Coshu.Coshv Cosh( u v ) Cosh( u v ) c ) Sinhu Sinh u Cosh u.sinhu Sinhu Sinh u Sol. a ) b ) c ) d ) d ).H.S..H.S. L.H.S. L.H.S. Sinh u Sinh v Cosh u Cosh v Sinh( u v ) Sinh( u v ) Sinhu.Coshv Coshu.Sinhv Sinhu.Coshv Coshu.Sinhv Sinhu.Coshv L.H.S. Cosh( u v ) Cosh( u v ) Coshu.Coshv Sinhu.Sinhv Coshu.Coshv Sinhu.Sinhv Coshu.Coshv L.H.S. Sinh( u u ) Sinhu.Coshu Coshu.Sinhu Sinhu.Coshu.Coshu ( Cosh u Sinh u ).Sinhu Sinhu.Cosh u Sinh u.h.s.( I ) Sinhu.( Sinh u ) Sinh u Sinhu Sinh u.h.s.( II ) Sinh u Sinh v Cosh u ( Cosh v ) Cosh u Cosh v.h.s. -5- Invrs hprbolic functions All hprbolic functions hav invrss that ar usful in intgration intrsting as diffrntiabl functions in thir own right. ٢١

22 ٢٢ Sinh Cosh ١-١ - ١ or Coth tanh ١ Sch Csch

23 ٢٣ Som usful idntitis Sinh ln Csch 6. Cosh ln Sch 5. tanh.ln Coth..ln tanh. ) ln( Cosh. ) ln( Sinh. EX- - riv th formula ) ln( Sinh Sol.- ) ln( or sinc nglctd ) ln( ithr. Sinh Sinh Lt

24 Problms. A bod of unknown tmpratur was placd in a room that was hld at o F. Aftr minuts, th bod's tmpratur was o F, minuts aftr th bod was placd in th room th bod's tmpratur 5 o F. Us Nwton's law of cooling to stimat th bod's initial tmpratur. (ans.- o F). A pan of warm watr 6 o C was put in a rfrigrator. Tn minuts latr, th watr's tmpratur was 9 o C, minuts aftr that, it was o C. Us Nwton's law of cooling to stimat how cold th rfrigrator was? (ans.- o C). Solv th following quations for valus of θ from -8 o to 8 o inclusiv i) tan θ + tan θ = ii) Cot θ= 5 Cos θ iii) Cos θ + Sc θ + 7 = iv) Cos θ + Sin θ + = (ans.i)-8,-5,,5,8; ii)-9,.5,9,68.5; iii)-9.5,9.5; iv)-9). Solv th following quations for valus of θ from o to 6 o inclusiv i) Cos θ Sin θ + = ii) tan θ = tan θ iii) Sin θ. Cos θ + Sin θ = iv) Cot θ + Cot θ = (ans.i)56.,.6,7; ii),,5,8,,,6; iii),9,5,7; iv)5,,5,) 5. If Sin θ = / 5, find without using tabls th valus of i) Cos θ ii) tan θ (ans. i) /5 ; ii) / ) 6. Find, without using tabls, th valus of Cos Sin, whn Cos is a) /8, b) 7/5, c) -9/ ( ans. a ), ;b ), ;c ), ) If Sin A = /5 Sin B = 5/, whr A B ar acut angls, find without using tabls, th valus of a) Sin(A+B), b) Cos(A+B), c) Cot(A+B) (ans. 56/65; /65; /56) 8. If tan A = -/7 tan B = /, whr A is obtus B is acut, find without using tabls th valu of A B. (ans. 5 ) 9.Prov th following idntitis ٢٤

25 i ) ii ) iii ) iv ) v ) vi ) vii ) viii ) i ) ) i ) ii ) iii ) iv ) v ) vi ) ) Sc Csc Sc.Csc Sin ( Sc ) Sc Cos Sin ( Sc tan ) Sin tan Cos Sc Sin Sc Sin Cos( A B ) Cos( A B ) tan B Sin( A B ) Sin( A B ) CosB CosA.Cos( A B ) SinA.Sin( A B ) tan A tan B tanc tan A.tan B.tanC tan( A B C ) tan B.tanC tanc.tan A tan A.tan B If A,B,C ar angls of a triangl,show that tan A tan B tanc tan A.tan B.tanC tan.sin h tan( h ) tan( h ) tan Cos Sin h Cos tan Cos Sin A Sin A tana Cos A Cos A Sin Cos ( Cos ) Sin A.Cos A Cos A.Sin A Sin A tan A tan A tana tan A Cos ( ) Cos Cot tan Cosh( u v ) Coshu.Coshv Sinhu.Sinhv ( Cosh Sinh ) thn vrif Cosh( u v ) Coshu.Coshv Sinhu.Sinhv vii ) Coshu.Sinhv Sinh( u v ) Sinh( u v ) viii ) Sinhu.Sinhv Cosh( u v ) Cosh( u v ) i ) Coshu Coshu Sinh u.coshu Cosh u Coshu n Coshn Sinhn Sin Sin. If u, prov that dduc formula for Sinθ, Cos u Cos Cosθ, tanθ in trms of u. (ans.(u -)/(u +); u/(u +);(u -)/(u +)) ٢٥

26 . If Sin( ) Cos( ) ; prov that tan tan. tan. If Sin( ) Cos( ) ; prov that tan.. If Cos Cos Sin Sin. Show that i ) Cos Cos Cos ii ) Sin Sin Sin. If Cos A.Cos B Cos, prov that Sin A.Cos B Cos A.Sin B Sin 5. If S = Sin θ C = Cos θ, simplif S.C S. S C S i), ii ), iii ) S C S C. C (ans.i) Sinθ; ii); iii) Scθ.Cscθ) 6. Eliminat θ from th following quations i) a.csc b.sc ii ) Sin Cos Sin Cos iii ) Sin tan Sin tan iv ) tan tan (ans. i ) a b ; ii) ; iii) ( ) ( ) ; iv) ) 7. In th acut angld triangl OPQ, th altitud O maks angls A B with OP OQ. Show b mans of aras that if OP=q, OQ=p, O=r p.q.sin(a+b) = q.r.sina + p.r. SinB. 8. Givn that Sin, find Cosα, tanα, Scα, Cscα. ( ans. ; ; ; ) 9. Evaluat th following prssions ٢٦

27 ) c ) Cot ( Cos ) a ) Sin( Cos b ) Csc ( Sc ) d ) Sin Sin ( ) f ) Cos ( Sin ) 6 ( ans. / ; / ; ; ;.6 ; / ) ) Cos( Sin.8 ). Find th angl α in th blow graph ( Hint α+β = 65o ) 65o ٢١ β ٥٠ α (ans..). Lt Sch u = /5, dtrmin th valus of th rmaining fiv hprbolic functions. ( ans. Coshu 5 / ; Sinhu / ; tanh u / 5 ; Cothu 5 / ; Cschu / ). writ th following prssions in trms of ponntials, writ th final rsult as simpl as ou can a ) Sinh(.ln ) b) Cosh Sinh c ) Cosh Sinh d ) ln( Cosh Sinh ) ln( Cosh Sinh ) (ans.(-)/(); ; -- ; ). Solv th quation for ; tanh = /5. (ans. ln ). Show that th distanc r from th origin O to th point P(Coshu,Sinhu) on th hprbola = is r Cosh u. Sinh = tan θ. Show that Cosh = Sc θ, tanh =Sin θ, Coth = Csc θ, Csch = Cot θ, Sch = Cos θ. 6. riv th formula tanh ln ; 7. Find lim Cosh ln. (ans. ln ) 5. If θ lis in th intrval ٢٧

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