高等数学重要公式手册

1/23ØIpf́lKŒ Òýps¹sûï„sûsinα=tanαcosαcosα=cotαsinαtanα=sinαsecαcotα=cosαcscαsecα=tanαcscαcscα=secαcotαpsûtanαcotα=1sinαcscα=1cosαsecα=1ôÒ ÒbABC-,ÒA„c&IŽÒA„ù¹Ôœ¹Y&IŽÒA„»¹Ôœ¹cIŽù¹Ô»¹ ÒýpRIØbl$Ҍî„ Òýpcos(α+β)=cosαcosβ-sinαsinβcos(α-β)=cosαcosβ+sinαsinβsin(α±β)=sinαcosβ±cosαsinβtan(α+β)=(tanα+tanβ)/(1-tanαtanβ)tan(α-β)=(tanα-tanβ)/(1+tanαtanβ)aaaaaa222222csc1cotsec1tan1cossin=+=+=+2/23 Ҍ„ Òýpsin(α+β+γ)=sinαcosβcosγ+cosαsinβcosγ+cosαcosβsinγ-sinαsinβsinγcos(α+β+γ)=cosαcosβcosγ-cosαsinβsinγ-sinαcosβsinγ-sinαsinβcosγtan(α+β+γ)=(tanα+tanβ+tanγ-tanαtanβtanγ)/(1-tanαtanβ-tanβtanγ-tanγtanα)…©ÒlAsinα+Bcosα=22sin()tABa++v-sint=22BAB+cost=22AAB+tant=BAAsinα+Bcosα=22cos()tABa-+tant=ABÒlsin(2α)=2sinα·cosα=2/(tanα+cotα)cos(2α)=cos^2(α)-sin^2(α)=2cos^2(α)-1=1-2sin^2(α)tan(2α)=2tanα/[1-tan^2(α)] Òlsin(3α)=3sinα-4sin^3(α)cos(3α)=4cos^3(α)-3cosαJÒlsin(α/2)=±1cos2a-cos(α/2)=±1cos2a+3/23tan(α/2)=±1cos1cosaa-+=sinα/(1+cosα)=(1-cosα)/sinαMBlsin^2(α)=(1-cos(2α))/2cos^2(α)=(1+cos(2α))/2tan^2(α)=(1-cos(2α))/(1+cos(2α))ýlsinα=2tan(α/2)/[1+tan^2(α/2)]cosα=[1-tan^2(α/2)]/[1+tan^2(α/2)]tanα=2tan(α/2)/[1-tan^2(α/2)]ïŒîlsinα·cosβ=(1/2)[sin(α+β)+sin(α-β)]cosα·sinβ=(1/2)[sin(α+β)-sin(α-β)]cosα·cosβ=(1/2)[cos(α+β)+cos(α-β)]sinα·sinβ=-(1/2)[cos(α+β)-cos(α-β)]Œîïlsinα+sinβ=2sin[(α+β)/2]cos[(α-β)/2]sinα-sinβ=2cos[(α+β)/2]sin[(α-β)/2]cosα+cosβ=2cos[(α+β)/2]cos[(α-β)/2]cosα-cosβ=-2sin[(α+β)/2]sin[(α-β)/2]¨ültanα+cotα=2/sin2αtanα-cotα=-2cot2α1+cos2α=2cos^2α4/231-cos2α=2sin^2α1+sinα=(sinα/2+cosα/2)^2vÖsinα+sin(α+2π/n)+sin(α+2π*2/n)+sin(α+2π*3/n)+……+sin[α+2π*(n-1)/n]=0cosα+cos(α+2π/n)+cos(α+2π*2/n)+cos(α+2π*3/n)+……+cos[α+2π*(n-1)/n]=0sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0üpl0C¢=1()xxmmm-¢=()()π(sin)sin2π(cos)cos2nnxxnxxn=+Łł=+Łłaxxaaaxxxxxxxxxxaxxln1)(logln)(cotcsc)(csctansec)(seccsc)(cotsec)(tan22=¢=¢-=¢=¢-=¢=¢222211)cotarc(11)(arctan11)(arccos11)(arcsinxxxxxxxx+-=¢+=¢--=¢-=¢5/23ú,ïhππ222002222222222222222222221sindcosddln()22dln22darcsin22darcsin(0)d2arctan()()()esnnnnxnIxxxxInxaxaxxaxxaCxaxaxxaxxaCxaxaxxaxCaaxxxaaxCaaxaxxaCaxbbxxabx--===+=+++++-=--+-+-=-+++=--+--=+---1inde(sincos)21ecosde(sincos)2xxxxxxxCxxxxC=-+=++6/23 Òýp„ ï22221d2d2tan11cos12sinuuxxuuuxuux+==+-=+=+–+=–+=+=+=+-=+=+-==+==CaxxaxxCxxxCxxxCaaxaCxxxxCxxxxCxxxxxCxxxxxxx)ln(dshdchchdshlndcscdcotcscsecdtanseccotdcscsindtandseccosd22222222CaxxaxCxaxaaxaxCaxaxaaxxCaxaxaxCxxxxCxxxxCxxxCxxx+=-+-+=-++-=-+=++-=++=+=+-=arcsindln21dln21darctan1dcotcsclndcsctanseclndsecsinlndcotcoslndtan222222227/23›Iýp$*́P Òýpl· Òýp9006004503000021 Òýp ÒýpÒaasincosaatancot212222232333331133100X(100X(xxxxxxxxxxxxxxxxxxxxxx-+=-+–=++=+-==+=-=----11ln21arth:)1ln(arch:1ln(arsh:eeeechshth:2eech:2eesh:22ÍÌòcÍÌòY& ÍÌòc&ÌòcÌòY&Ìòc&...590457182818284.2e)11(lim1sinlim0==+=¥fifixxxxxx8/23·ñülýpÒAsincostancot-α-sinαcosα-tanα-cotα90°-αcosαsinαcotαtanα90°+αcosα-sinα-cotα-tanα180°-αsinα-cosα-tanα-ctgα180°+α-sinα-cosαtanαcotα270°-α-cosα-sinαcotαtanα270°+α-cosαsinα-cotα-tanα360°-α-sinαcosα-tanα-cotα360°+αsinαcosαtanαcot᷌îÒl·Œîïl Òýp„Ò¦b—l¾α:ûÒȹø„Ò„ Òýp„øIsin2kπα sinαcos2kπα cosαtan2kπα tanαcot2kπα cotαlŒ¾α:ûÒπ+α„ Òýpα„ ÒýpKô„sû2sin2sin2coscos2cos2cos2coscos2sin2cos2sinsin2cos2sin2sinsinbabababababababababababa-+=--+=+-+=--+=+abbababababababababababacotcot1cotcot)cot(tantan1tantan)tan(sinsincoscos)cos(sincoscossin)sin(–=––=–=––=–mmm9/23sinπα sinαcosπα cosαtanπα tanαcotπα cotαl ûÒα-α„ ÒýpKô„sûsinα sinαcosα cosαtanα tanαcotα cotαlÛ)(lŒŒl ïå—0π-αα„ ÒýpKô„sûsinπα sinαcosπα cosαtanπα tanαcotπα cotαl”)(lŒl ïå—02π-αα„ ÒýpKô„sûsin2πα sinαcos2πα cosαtan2πα tanαcot2πα cotαlmπ/2±αÊ3π/2±αα„ ÒýpKô„sûsinπ/2α cosαcosπ/2α sinαtanπ/2α cotαcotπ/2α tanαsinπ/2α cosαcosπ/2α sinαtanπ/2α cotαcotπ/2α tanαsin3π/2α cosαcos3π/2α sinαtan3π/2α cotαcot3π/2α tanαsin3π/2α cosαcos3π/2α sinαtan3π/2α cotαcot3π/2α tanα(åkZ)10/23·Òl·JÒlaaaaaaaaaaaaaaaaaacos1sinsincos1cos1cos12cotcos1sinsincos1cos1cos12tan2cos12cos2cos12sin-=+=-+–=+=-=+-–=+–=-–=·c&šRCcBbAa2sinsinsin===·Y&šCabbaccos2222-+=·Í Òýp'(xxxxcotarc2πarctanarccos2πarcsin-=-=Ø6üpl±(l)()()()2()1()(0)()()(!)1()1(!2)1(C)(nkknnnnnkkknknnuvvukknnnvunnvnuvuvuuv+++--++¢¢-+¢+==---=-LLLaaaaaaaaaa2333tan31tantan33tancos3cos43cossin4sin33sin--=-=-=aaaaaaaaaaaaaa222222tan1tan22tancot21cot2cotsincossin211cos22coscossin22sin-=-=-=-=-==11/23-šüp”(.)(F)()()()()()())(()()(Éå-šöï-š1/Sï-šÉå-šxxFfaFbFafbfabfafbf=¢¢=---¢=-xxxò‡.1.0.)1(ddlimM.sMM:.tan,1d3202aKaKyyssKMMsKydxyss==¢+¢¢==DD=¢D¢DDD==¢¢+=fiD„J„:ô¿¹„ò‡'Ϲ¿œ‡„Òع0ÎsGò‡v-'®laaaaašï„Ñ¡—)](4)(2)[(3d)()(21d)()(d)(1312420110110----+++++++++-»œßøŒºØ++++-»+++-»nnnbannbanbayyyyyyyynabxxfyyyynabxxfyyynabxxfLLLL›i¿Õ¯bÕébÕšï”(øsl--====babattfabxxfabykrmmkFApFsFWd)(1d)(1,2221G¹9ýp„sG:›ûp›4‹›Ÿ12/23®¹„øs‚õ.)(dd)