Edexcel FP3. Hyperbolic Functions. PhysicsAndMathsTutor.com

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1 Eeel FP Hpeoli Futios PhsisAMthsTuto.om

2 . Solve the equtio Leve lk 7seh th 5 Give ou swes i the fom l whee is tiol ume. 5 7 Sih 5 Cosh osh 7 Sih 5osh's 7 Ee 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 osh i tems of epoetils, pove tht osh sih Leve lk Solve the equtio osh sih 5, givig ou swes s et logithms. I si ti I t Ee e I t e e I t e e t I I t t e e K osh 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 uve C hs equtio sih, the uve C hs equtio e. Leve lk Sketh the gph of the uves C C o oe set of es, givig the equtio of smptote the ooites of poits whee the uves oss 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 7. f 5osh 4sih, R Show tht f e 9e Hee Leve lk solve f 5 show tht l l 5osh 4sih π 8 f 5os.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 *P40A0*

6 Questio 7 otiue f e I fi 5oh lis I I Is Zz ELIA si *P40A0* ti'ed t l Ti't ELE I 8 Is Let u e IT.se u Leve lk e o Tu ove

7 7. Leve lk O Figue The uves show i Figue hve equtios 6 osh 9 sih Usig the efiitios of sih osh i tems of e, fi et vlues fo the -ooites of the two poits whee the uves iteset. 6 The fiite egio etwee the two uves is show she i Figue. Usig lulus, fi the e of the she egio, givig ou swe i the fom l, whee, e iteges. 6 Iteset 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 4 *P4956A04*

8 Questio 7 otiue I Ae 9 Sih Gosh 94 I't 9 Zeoshoe 6 sih Iz Iz 9k 9 I 7 t 9A 7 94 t I 4 It 4 Ig 4 Leve lk *P4956A05* 5 Tu ove

9 0 Eeel AS/A level Mthemtis Fomule List: Futhe Pue Mthemtis FP Issue Septeme 009 Futhe Pue Mthemtis FP Cites sittig FP m lso equie those fomule liste ue Futhe Pue Mthemtis FP, Coe Mthemtis C C4. Vetos The esolve pt of i the ietio of is. The poit iviig AB i the tio µ λ : is µ λ λ µ Veto pout: ˆ si k j i θ... If A is the poit with positio veto k j i the ietio veto is give k j i, the the stight lie though A with ietio veto hs tesi equtio λ z The ple though A with oml veto k j i hs tesi equtio. z 0 whee The ple though o-ollie poits A, B C hs veto equtio µ λ µ λ µ λ The ple though the poit with positio veto pllel to hs equtio t s The pepeiul iste of,, γ β α fom 0 z is γ β α.

10 Hpeoli futios osh sih sih sih osh osh osh sih osh l{ } sih l{ } th l < Cois Ellipse Pol Hpeol Retgul Hpeol St Fom 4 Pmeti Fom osθ, siθ t, t se θ, t θ ± osh θ, sih θ t, t Eetiit e < e e e > e e Foi ± e, 0, 0 ± e, 0 ±, ± Dieties ± e ± ± e Asmptotes oe oe ± 0, 0 Eeel AS/A level Mthemtis Fomule List: Futhe Pue Mthemtis FP Issue Septeme 009

11 Diffeetitio f f si os t sih osh osh sih th seh sih osh th Itegtio ostt; > 0 whee elevt f f sih osh osh sih th l osh si t < osh, l{ } sih, l l l { } th > < Eeel AS/A level Mthemtis Fomule List: Futhe Pue Mthemtis FP Issue Septeme 009

12 Eeel AS/A level Mthemtis Fomule List: Futhe Pue Mthemtis FP Issue Septeme 009 A legth s tesi ooites t t t s pmeti fom Sufe e of evolutio S s π π t t t π

13 Futhe Pue Mthemtis FP Cites sittig FP m lso equie those fomule liste ue Coe Mthemtis C C. Summtios 6 4 Numeil solutio of equtios The Newto-Rphso itetio fo solvig f 0 : f f Cois Pol Retgul Hpeol St Fom 4 Pmeti Fom t, t t, t Foi, 0 Not equie Dieties Not equie Mti tsfomtios Atilokwise ottio though θ out O: osθ siθ siθ osθ Refletio i the lie os θ si θ tθ : si θ os θ I FP, θ will e multiple of Eeel AS/A level Mthemtis Fomule List: Futhe Pue Mthemtis FP Issue Septeme 009

14 Coe Mthemtis C4 Cites sittig C4 m lso equie those fomule liste ue Coe Mthemtis C, C C. Itegtio ostt f f se k t k k t l se ot l si ose l ose ot, l t se l se t, l t 4 π v u u uv v Eeel AS/A level Mthemtis Fomule List: Coe Mthemtis C4 Issue Septeme 009 7

15 Coe Mthemtis C Cites sittig C m lso equie those fomule liste ue Coe Mthemtis C C. Logithms epoetils e l Tigoometi ietities si A ± B si Aos B ± os Asi B os A ± B os Aos 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 os A B A B si A si B os si A B A B os A os B os os A B A B os A os B si si π Diffeetitio f t k se ot ose f g f k se k se t ose ose ot f g f g g 6 Eeel AS/A level Mthemtis Fomule List: Coe Mthemtis C Issue Septeme 009

16 Eeel AS/A level Mthemtis Fomule List: Coe Mthemtis C Issue Septeme Coe Mthemtis C Cites sittig C m lso equie those fomule liste ue Coe Mthemtis C. Cosie ule os A Biomil seies whee!!! C <, Logithms epoetils log log log Geometi seies u S S fo < Numeil itegtio The tpezium ule: h{ 0... }, whee h

17 Coe Mthemtis C Mesutio Sufe e of sphee 4π Ae of uve sufe of oe π slt height Aithmeti seies u S l [ ] 4 Eeel AS/A level Mthemtis Fomule List: Coe Mthemtis C Issue Septeme 009

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Edexcel FP3. Hyperbolic Functions. PhysicsAndMathsTutor.com

Edexcel FP3. Hyperbolic Functions. PhysicsAndMathsTutor.com Eecel FP Hpeolic Fuctios PhsicsAMthsTuto.com . 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