Advanced Higher Maths: Formulae

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Jessica Carr
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1 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 fomule, s it is possile to deive them fom sctch i the em. ut 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 out memoisig these. Just use this sheet to help jog you memoy i clsswok d homewok. Oe o two of these fomule e o the syllus, ut e sufficietly oscue tht I do t thik it essetil to memoise them. Essetil Tigoometic Idetities: (fom Itemedite d Highe) Liks etwee tios Essetil Fomule to kow off y het fo the em (G) cos Asi A Othe useful oes tht my e useful fo homewok/clsswok etc. t A sec A (A) cot A cosec A (R) si A t A cos A Squed cos ( cos ) si ( cos ) Compoud si( A ) si Acos cos Asi t A t t( A ) (R) Agle cos( A ) cos Acos si Asi t A t Doule si( A) si Acos A ta Agle (A) t( A) (R) cos( A) cos Asi A t A Ect Vlues(you should kow ll these, though thee is o o-clculto ppe, ulike Highe) si cos t udef. 0 udef. 0 Uit.3: Comple Numes Fo the comple ume, z i, the modulus is give y z d the gumet is give y t The cojugte is z i Negtive fcts: si( ) si( ) cos( ) cos( ) t( ) t( ) De Moive s Theoem sys tht fo y z (cos isi ), the z (cos isi ) ( ) Newttle Commuity High School D Wtkis 0

2 Advced Highe Mthemtics Uits. d.: Diffeetitio f( ) f '( ) t sec sec sec t cosec cosec cot cot cosec l f( ) f '( ) f( ) f ( ) f '( ) - si - - cos - - t To diffeetite ivese fuctio: d (A) d Pmetic Equtios (whee f(), t y g() t ): Gdiet (diectio of movemet) = d Speed = dt dt d y y y 3 d Uits.3 d.: Itegtio d dt d dt (G) Essetil Itegls to Le f ( ) f ( d ) sec t C t l sec C f '( ) f ( ) l f ( ) C t si - - C C (A) Could use sustitutio if eeded: f ( ) f ( ) d si t C C (R) To sve you time i hd questios fo homewok/clsswok, o eed to memoise: f ( ) f ( ) d cosec l cosec cot C cot l si C sec l sec t C Volume of solid of evolutio f() etwee d : Aout is: V f( ) d Aout y is: V f( y) Uit.4: Popeties of fuctios Odd fuctio: f ( ) f( ) Eve fuctio: f ( ) f( ) (80 ottiol symmety) (lie symmety out the y-is) Newttle Commuity High School D Wtkis 0

3 Advced Highe Mthemtics 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: 3... d (...) (A) d lso lim( ) e Impott Idetities k ( ) k k k ( )( ) (ote: this is med specificlly o syllus) (A) 6 k 3 k k 4 ( ) k k (lso med specificlly o syllus) (A) 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... 3! 5! 7! 4 6 cos...! 4! 6! O syllus ut less essetil (R): t l( ) Newttle Commuity High School D Wtkis 0

4 Advced Highe Mthemtics Uit.: iomil Theoem The coefficiet of the th tem i the iomil epsio ( y) is C!!( )! y Uit 3.: Vectos, Lies d Ples Agle etwee two vectos: (Highe) cos Equtios of 3d lie: though (, y, z ) d with diectio vecto d ijck Pmetic fom Symmetic fom t y y z z y y t ( td ) ( t) c z z ct Equtios of ple: l Noml is m Poit o lie = P (with positio vecto ) Vecto equtio Symmetic/Ctesi Pmetic (A) l my z k c whee k ( d c e y two opllel vectos i ple) Agle etwee two lies = Acute gle etwee thei diectio vectos Agle etwee two ples = Acute gle etwee thei omls Agle etwee lie d ple = 90 (Acute gle etwee d d) i j k Coss (vecto) poduct: 3 3 Scl tiple poduct: 3 ( c) (A) 3 c c c 3 (this is impott ut is oly ted me, s you would t hve to ememe it, s y em questio would hve to tell you ( c )) Newttle Commuity High School D Wtkis 0

5 Advced Highe Mthemtics Uit 3.: Mtices mtices 3 3 mtices A c d c A d e f g h i Detemit d Ivese d det A d c d A d c c e f d f d e det A c h i g i g h ( A) A ( A) T T A T det A det Adet (A) Tsfomtio Mtices 0 Reflectio i -is 0, Reflectio i y-is Elgemet y scle fcto 0, Rottio y θ degees cos si si cos Uit 3.4: Diffeetil Equtios (G) ( ) Fo Py ( ) Q ( ) d, the Itegtig Fcto I() is d the solutio is give y I( y ) IQd ( ) ( ) e P d Secod Ode Diffeetil Equtios COMPLEMENTARY FUNCTION (Homogeeous Equtios) Ntue of oots Fom of geel solutio Two distict el m d m y Ae e Rel d equl m m y Ae m e Comple cojugte m p iq y e p ( Acosq si q) PARTICULAR INTEGRAL (Ihomogeeous Equtios) Right-hd side cotis Fo Pticul Itegl, ty si o cos y Pcos Qsi e y Pe Lie epessio y y P Q Qudtic epessio y c y P Q R Newttle Commuity High School D Wtkis 0

Advanced Higher Maths: Formulae

Advanced Higher Maths: Formulae : 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