Advanced Higher Maths: Formulae

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Marsha Greer
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1 : Fomule Gee (G): Fomule you bsolutely must memoise i ode to pss Advced Highe mths. Remembe you get o fomul sheet t ll i the em! Ambe (A): You do t hve to memoise these fomule, s it is possible to deive them fom sctch i the em. But it will sve you lot of time if you do choose to memoise them, d I dvise tht you do. Red (R): Do t woy bout memoisig these. Just use this sheet to help jog you memoy i clsswok d homewok. Oe o two of these fomule e o the syllbus, but e sufficietly obscue tht I do t thik it essetil to memoise them. Essetil Tigoometic Idetities: (fom Itemedite d Highe) Liks betwee tios Essetil Fomule to kow by het fo the em (G) cos A + si A = si A t A = cos A Compoud si( A ± B) = si Acos B ± cos Asi B Agle cos( A ± B) = cos Acos B si Asi B Double si( A) = si Acos A Agle cos( A) = cos A si A Squed cos = ( + cos ) si = ( cos ) Othe useful oes tht my be useful fo homewok/clsswok etc. + t A = sec A cot A + = cosec A (A) t A ± t B t( A ± B ) = (R) t A t B t A t( A ) = (R) t A Uit.: Biomil Theoem The coefficiet of the th tem i the biomil epsio ( + y ) is C! = =!( )! y Uit.3: Comple Numbes Fo the comple umbe, z = + bi, the modulus is give by z = + b b d the gumet is give by t θ = π < θ < π De Moive s Theoem sys tht fo y z = (cosθ + i si θ ), the z = (cos θ + isi θ ) Newbttle Commuity High School D Wtkis 0

2 Uits. d.: Diffeetitio f f '( ) t sec sec sec t cosec cosec cot cot cosec l f ( ) f '( ) f ( ) To diffeetite ivese fuctio: f f '( ) si cos t d f ( ) = (A) d f '( f ( )) + Pmetic Equtios (whee = f ( t), y = g( t ) ): Gdiet (diectio of movemet) = d Speed = + d y d = d d d o d = d d y y y d = 3 (A) Uits.3 d.: Itegtio (G) Essetil Itegls to Le f f d sec t t l sec f '( ) f ( ) + l f t si (A)Could use substitutio if eeded: f f d + si t (R) To sve you time i hd questios fo homewok/clsswok, o eed to memoise: f f d cosec l cosec + cot cot l si sec l sec + t Volume of solid of evolutio f() bout is: b V π f d = Newbttle Commuity High School D Wtkis 0

3 Uit.4: Sequeces d Seies Aithmetic Seies Geometic Seies u = + ( ) d u = ( ) S = ( + ( ) d) S = S = < I pticul, you e supposed to kow tht s cosequece of the lst fomul (A): 3... = (...) + b = + b b d d lso lim( + ) = e (lso metioed specificlly o syllbus) (R) Impott Idetities k = k = k = k = ( + ) k = ( + )( + ) 6 k = ( + ) 3 4 (lso ote: this is the sme s k = = (ote: this is med specificlly o syllbus) (A) (ote: this is med specificlly o syllbus) (A) k = k : elisig this my help memoise it) Uit 3.3: Mclui Seies ( ) f (0) f (0) f = f (0) + f (0) (G)!! d i pticul: Vey useful to memoise (A): 3 e = ! 3!! si = ! 5! 7! 4 6 cos = ! 4! 6! O syllbus but less essetil (R): t = l( + ) = Newbttle Commuity High School D Wtkis 0

4 Uit 3.: Vectos, Lies d Ples (G) Agle betwee two vectos: (Highe) b = b cosθ Equtios of lie: Pmetic fom = + tl y = b + tm ( = + t d ) z = c + t Symmetic/Ctesi fom y b z c = = = t l m Equtios of ple: Noml is l m Poit o lie = P (with positio vecto ) Vecto equtio Symmetic/Ctesi Pmetic = l + my + z = k = + µ b + λ c o ( ) = 0 whee k = (b d c e y two o pllel vectos i ple) Agle betwee two ples = Agle betwee thei omls Agle betwee lie d ple = (Agle betwee d d) 90 i j k Coss poduct: b = 3 b b b 3 Scl tiple poduct: i( b c ) = 3 b b b 3 c c c 3 Newbttle Commuity High School D Wtkis 0

5 Uit 3.: Mtices (G) mtices 3 3 mtices b A = c d b c A = d e f g h i Detemit d Ivese d b det A = d bc d A = d bc c e f d f d e det A = b + c h i g i g h ( AB) = B A Tsfomtio Mtices 0 Reflectio i is 0, Reflectio i y is Elgemet by scle fcto 0, Rottio by θ degees cosθ si θ siθ cos θ Uit 3.4: Diffeetil Equtios (G) Fo P y Q( ) d + =, the Itegtig Fcto I() is d the solutio is give by I y = I Q d e P d Secod Ode Diffeetil Equtios Ntue of oots Fom of geel solutio Two distict el m d m y = Ae + Be Rel d equl m m y = Ae m + Be Comple cojugte m = p ± iq y = e p ( Acos q + B si q ) Newbttle Commuity High School D Wtkis 0

Advanced Higher Maths: Formulae

Advanced Higher Maths: Formulae Advced Highe Mths: Fomule Advced Highe Mthemtics Gee (G): Fomule you solutely must memoise i ode to pss Advced Highe mths. Rememe you get o fomul sheet t ll i the em! Ame (A): You do t hve to memoise these