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1 Eecel FP Hpeolic Fuctios PhsicsAMthsTuto.com

2 . Solve the equtio Leve lk 7sech th 5 Give ou swes i the fom l whee is tiol ume. 5 7 Sih 5 Cosh cosh c 7 Sih 5cosh's 7 Ece e I E e e 4 e te 5e 55 O 5e 55 te e 4 O Ge e t 5 7 O 7 e t O e Ike e t I e e f I t I I I o I *M545A08*

3 . Sttig fom the efiitios of sih cosh i tems of epoetils, pove tht cosh sih Leve lk Solve the equtio cosh sih 5, givig ou swes s ect logithms. I si ti I t Ece e I t e e I t e e t I I t t e e K cosh sih5 I t Sih sih 5 Sih Sih Sih 4 0 Sih 7 si.h so Iz o Sih sih Z o se sihf I Ztf o I 55 I *N589RA068*

4 5. The cuve C hs equtio sih, the cuve C hs equtio e. Leve lk Sketch the gph of the cuves C C o oe set of es, givig the equtio of smptote the cooites of poits whee the cuves coss the es. 4 Solve the equtio sih e, givig ou swe i the fom l k, whee k is itege. 5 µ o p o 7 Smh Je o e I flk e o 7 si e e EZ e 9e e e 6 Ge e K to 9e 0 o oots EZ 0 e I *P544A08* zl

5 Questio 5 cotiue f 5cos.hu 4Sih Ele te q e e Ee t e Iz e t 95 f 5 e t 9 e 7 5 e t 9 e to t 9 toe e toe 9 0 e e 9 o e o e 9 0 O I 9 Leve lk *P544A08* Tu ove

6 7. f 5cosh 4sih, R c Show tht f e 9e Hece solve f 5 c show tht l l fi 5cosh 4sih π 8 I fi 5coh lis I Is Zz ELIA si *P40A0* e ti'ed t l Ti't ELE I 8 Is Let u 4 5 e IT.se u Leve lk e oc

7 Questio 7 cotiue Itesect whe g 6E e't e 9 E e e e t e 9 e te e t e 9 to 4 9 e t 0 4 e e to 4 e o e I I I *P40A0* Leve lk Tu ove

8 7. I Leve lk Ae 9 Sih Gosh I't O 9 Zeoshoe 6 sih Figue The cuves show i Figue hve equtios It cosh 9 sih Iz Iz 9k Usig the efiitios of sih cosh i tems of e, fi ect vlues fo the -cooites of the two poits whee the cuves itesect. 6 9 I 7 t 9A 7 The fiite egio etwee the two cuves is show she i Figue. Usig clculus, fi the e of the she egio, givig ou swe i the fom l c, whee, c e iteges. 94 t I 4 Ig 4 4 *P4956A04*

9 Questio 7 cotiue Leve lk *P4956A05* 5 Tu ove

10 0 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009 Futhe Pue Mthemtics FP Cites sittig FP m lso equie those fomule liste ue Futhe Pue Mthemtics FP, Coe Mthemtics C C4. Vectos The esolve pt of i the iectio of is. The poit iviig AB i the tio µ λ : is µ λ λ µ Vecto pouct: ˆ si k j i θ c. c. c. c c c If A is the poit with positio vecto k j i the iectio vecto is give k j i, the the stight lie though A with iectio vecto hs ctesi equtio λ z The ple though A with oml vecto k j i hs ctesi equtio. z 0 whee The ple though o-collie poits A, B C hs vecto equtio c c µ λ µ λ µ λ The ple though the poit with positio vecto pllel to c hs equtio c t s The pepeicul istce of,, γ β α fom 0 z is γ β α.

11 Hpeolic fuctios cosh sih sih sih cosh cosh cosh sih cosh l{ } sih l{ } th l < Coics Ellipse Pol Hpeol Rectgul Hpeol St Fom 4 c Pmetic Fom cosθ, siθ t, t sec θ, t θ ± cosh θ, sih θ ct, c t Ecceticit e < e e e > e e Foci ± e, 0, 0 ± e, 0 ± c, ± c Diectices ± e ± ± c e Asmptotes oe oe ± 0, 0 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009

12 Diffeetitio f f csi ccos ct sih cosh cosh sih th sech sih cosh th Itegtio costt; > 0 whee elevt f f sih cosh cosh sih th l cosh csi ct < cosh, l{ } sih, l l l { } th > < Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009

13 Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009 Ac legth s ctesi cooites t t t s pmetic fom Sufce e of evolutio S s π π t t t π

14 Futhe Pue Mthemtics FP Cites sittig FP m lso equie those fomule liste ue Coe Mthemtics C C. Summtios 6 4 Numeicl solutio of equtios The Newto-Rphso itetio fo solvig f 0 : f f Coics Pol Rectgul Hpeol St Fom 4 c Pmetic Fom t, t ct, c t Foci, 0 Not equie Diectices Not equie Mti tsfomtios Aticlockwise ottio though θ out O: cosθ siθ siθ cosθ Reflectio i the lie cos θ si θ tθ : si θ cos θ I FP, θ will e multiple of Eecel AS/A level Mthemtics Fomule List: Futhe Pue Mthemtics FP Issue Septeme 009

15 Coe Mthemtics C4 Cites sittig C4 m lso equie those fomule liste ue Coe Mthemtics C, C C. Itegtio costt f f sec k t k k t l sec cot l si cosec l cosec cot, l t sec l sec t, l t 4 π v u u uv v Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C4 Issue Septeme 009 7

16 Coe Mthemtics C Cites sittig C m lso equie those fomule liste ue Coe Mthemtics C C. Logithms epoetils e l Tigoometic ietities si A ± B si Acos B ± cos Asi B cos A ± B cos Acos B si Asi B t A ± t B t A ± B A ± B k t A t B A B A B si A si B si cos A B A B si A si B cos si A B A B cos A cos B cos cos A B A B cos A cos B si si π Diffeetitio f t k sec cot cosec f g f k sec k sec t cosec cosec cot f g f g g 6 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 009

17 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme Coe Mthemtics C Cites sittig C m lso equie those fomule liste ue Coe Mthemtics C. Cosie ule c c cos A Biomil seies whee!!! C <, Logithms epoetils log log log Geometic seies u S S fo < Numeicl itegtio The tpezium ule: h{ 0... }, whee h

18 Coe Mthemtics C Mesutio Sufce e of sphee 4π Ae of cuve sufce of coe π slt height Aithmetic seies u S l [ ] 4 Eecel AS/A level Mthemtics Fomule List: Coe Mthemtics C Issue Septeme 009

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PhysicsAndMathsTutor.com PhysicsAMthsTuto.com . Leve lk A O c C B Figue The poits A, B C hve positio vectos, c espectively, eltive to fie oigi O, s show i Figue. It is give tht i j, i j k c i j k. Clculte () c, ().( c), (c) the