HYPERLINK"http://www.521yy.com/tools/maths/"三角函数(Trigonometric)是
数学
数学高考答题卡模板高考数学答题卡模板三年级数学混合运算测试卷数学作业设计案例新人教版八年级上数学教学计划
中属于初等函数中的超越函数的一类函数。它们的本质是任意角的集合与一个比值的集合的变量之间的映射。通常的三角函数是在平面直角坐标系中定义的,其定义域为整个实数域。另一种定义是在直角三角形中,但并不完全。现代数学把它们描述成无穷数列的极限和微分方程的解,将其定义扩展到复数系。它包含六种基本函数:正弦、余弦、正切、余切、正割、余割。由于三角函数的周期性,它并不具有单值函数意义上的反函数。三角函数在复数中有较为重要的应用。在物理学中,三角函数也是常用的工具。三角函数起源 “三角学”,英文Trigonometry,法文Trigonometrie,德文Trigonometrie,都来自拉丁文Trigonometria。现代三角学一词最初见于希腊文。最先使用Trigonometry这个词的是皮蒂斯楚斯(BartholomeoPitiscus,1516-1613),他在1595年出版一本著作《三角学:解三角学的简明处理》,创造了这个新词。它是由τριγωυου(三角学)及μετρειυ(测量)两字构成的,原意为三角形的测量,或者说解三角形。古希腊文里没有这个字,原因是当时三角学还没有形成一门独立的科学,而是依附于天文学。因此解三角形构成了古代三角学的实用基础。 早期的解三角形是因天文观测的需要而引起的。还在很早的时候,由于垦殖和畜牧的需要,人们就开始作长途迁移;后来,贸易的发展和求知的欲望,又推动他们去长途旅行。在当时,这种迁移和旅行是一种冒险的行动。人们穿越无边无际、荒无人烟的草地和原始森林,或者经水路沿着海岸线作长途航行,无论是那种方式,都首先要明确方向。那时,人们白天拿太阳作路标,夜里则以星星为指路灯。太阳和星星给长期跋山涉水的商队指出了正确的道路,也给那些沿着遥远的异域海岸航行的人指出了正确方向。 就这样,最初的以太阳和星星为目标的天文观测,以及为这种观测服务的原始的三角测量就应运而生了。因此可以说,三角学是紧密地同天文学相联系而迈出自己发展史的第一步的同角HYPERLINK"http://www.521yy.com/tools/maths/"三角函数的基本关系式倒数关系:商的关系:平方关系:tanα·cotα=1sinα·cscα=1cosα·secα=1sinα/cosα=tanα=secα/cscαcosα/sinα=cotα=cscα/secαsin2α+cos2α=11+tan2α=sec2α1+cot2α=csc2α 诱导公式sin(-α)=-sinαcos(-α)=cosαtan(-α)=-tanαcot(-α)=-cotα sin(π/2-α)=cosαcos(π/2-α)=sinαtan(π/2-α)=cotαcot(π/2-α)=tanαsin(π/2+α)=cosαcos(π/2+α)=-sinαtan(π/2+α)=-cotαcot(π/2+α)=-tanαsin(π-α)=sinαcos(π-α)=-cosαtan(π-α)=-tanαcot(π-α)=-cotαsin(π+α)=-sinαcos(π+α)=-cosαtan(π+α)=tanαcot(π+α)=cotαsin(3π/2-α)=-cosαcos(3π/2-α)=-sinαtan(3π/2-α)=cotαcot(3π/2-α)=tanαsin(3π/2+α)=-cosαcos(3π/2+α)=sinαtan(3π/2+α)=-cotαcot(3π/2+α)=-tanαsin(2π-α)=-sinαcos(2π-α)=cosαtan(2π-α)=-tanαcot(2π-α)=-cotαsin(2kπ+α)=sinαcos(2kπ+α)=cosαtan(2kπ+α)=tanαcot(2kπ+α)=cotα(其中k∈Z) 两角和与差的HYPERLINK"http://www.521yy.com/tools/maths/"三角函数公式万能公式sin(α+β)=sinαcosβ+cosαsinβsin(α-β)=sinαcosβ-cosαsinβcos(α+β)=cosαcosβ-sinαsinβcos(α-β)=cosαcosβ+sinαsinβ tanα+tanβtan(α+β)=—————— 1-tanα·tanβ tanα-tanβtan(α-β)=—————— 1+tanα·tanβ 2tan(α/2)sinα=—————— 1+tan2(α/2) 1-tan2(α/2)cosα=—————— 1+tan2(α/2) 2tan(α/2)tanα=—————— 1-tan2(α/2) 半角的正弦、余弦和正切公式HYPERLINK"http://www.521yy.com/tools/maths/"\t"_blank"三角函数的降幂公式 二倍角的正弦、余弦和正切公式三倍角的正弦、余弦和正切公式sin2α=2sinαcosαcos2α=cos2α-sin2α=2cos2α-1=1-2sin2α 2tanαtan2α=————— 1-tan2αsin3α=3sinα-4sin3αcos3α=4cos3α-3cosα 3tanα-tan3αtan3α=—————— 1-3tan2α 三角函数的和差化积公式三角函数的积化和差公式 α+β α-βsinα+sinβ=2sin—--·cos—-— 2 2 α+β α-βsinα-sinβ=2cos—--·sin—-— 2 2 α+β α-βcosα+cosβ=2cos—--·cos—-— 2 2 α+β α-βcosα-cosβ=-2sin—--·sin—-— 2 2 1sinα·cosβ=-[sin(α+β)+sin(α-β)] 2 1cosα·sinβ=-[sin(α+β)-sin(α-β)] 2 1cosα·cosβ=-[cos(α+β)+cos(α-β)] 2 1sinα·sinβ=--[cos(α+β)-cos(α-β)] 2 化asinα±bcosα为一个角的一个三角函数的形式(辅助角的三角函数的公式)三角函数值-应用 三角函数在复数中有较为重要的应用。在物理学中,三角函数也是常用的工具。 它有六种基本函数: 函数名正弦余弦正切余切正割余割 符号sincostancotseccsc 正弦函数sin(A)=a/c 余弦函数cos(A)=b/c 正切函数tan(A)=a/b 余切函数cot(A)=b/a 其中a为对边,b为邻边,c为斜边HYPERLINK"http://www.baike.com/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E5%80%BC"\l"#"\t"_self"三角函数值-三角函数值表 sin0=sin0°=0 cos0=cos0°=1 tan0=tan0°=0 sin15=0.65028784015711686582974027098887;sin15°=(√6-√2)/4 cos15=-0.75968791285882127384814640363328;cos15°=(√6+√2)/4 tan15=-0.85599340090851876402619277823803;tan15°=2-√3 sin30=-0.98803162409286178998774890729446;sin30°=1/2 cos30=0.15425144988758405071866214661421;cos30°=√3/2 tan30=-6.405331196646275784896075505668;tan30°=√3/3 sin45=0.85090352453411842486237967761804;sin45°=√2/2 cos45=0.52532198881772969604746440482356;cos45°=sin45°=√2/2 tan45=1.6197751905438615499827965173949;tan45°=1 sin60=-0.30481062110221670562564946547843;sin60°=√3/2 cos60=-0.95241298041515629269381659599293;cos60°=1/2 tan60=0.32004038937956297493818761970571;tan60°=√3 sin75=-0.38778163540943043773094219282288;sin75°=cos15° cos75=0.92175126972474931639229684521414;cos75°=sin15° tan75=-0.42070095062112435384419621180484;tan75°=sin75°/cos75°=2+√3 sin90=0.89399666360055789051826949840421;sin90°=cos0°=1 cos90=-0.44807361612917015236547731439964;cos90°=sin0°=0 tan90=-1.9952004122082420252873530763796;tan90°不存在 sin105=-0.9705352835374847209338032056452;sin105°=cos15° cos105=-0.24095904923620141814219566395281;cos105°=-sin15° tan105=4.0278017638844193492341772197301;tan105°=-cot15° sin120=0.58061118421231428928240695979069;sin120°=cos30° cos120=0.81418097052656176790578986177889;cos120°=-sin30° tan120=0.71312300978590911614556745464875;tan120°=-tan60° sin135=0.088368686104001433036970666416798;sin135°=sin45° cos135=-0.99608783514118495835764440142782;cos135°=-cos45° tan135=-0.088715756770060446136984268003075;tan135°=-tan45° sin150=-0.71487642962916463143638609739663;sin150°=sin30° cos150=0.69925080647837513141645161882553;cos150°=-cos30° tan150=-1.0223462354365875649863661852619;tan150°=-tan30° sin165=0.99779727944989076515961268295525;sin165°=sin15° cos165=-0.066336936335623731837538732255201;cos165°=-cos15° tan165=-15.041353046538895371867886988769;tan165°=-tan15° sin180=-0.80115263573383047774673111582099;sin180°=sin0°=0 cos180=-0.5984600690578581389679486481597;cos180°=-cos0°=-1 tan180=1.3386902103511543616808987449579;tan180°=0 sin195=0.21945466799406361597859718309894;sin195°=-sin15° cos195=0.97562269791944432311811100484535;cos195°=-cos15° tan195=0.22493805080802215296340937063051;tan195°=tan15° sin360=0.95891572341430650775887594775378;sin360°=sin0°=0 cos360=-0.28369109148652733463742432444589;cos360°=cos0°=1 tan360=-3.3801404139609579824775356112668;tan360°=tan0°=0 cos72=[(√5)-1]/4(利用黄金等腰三角形可得出) sin1=0.01745240643728351sin2=0.03489949670250097sin3=0.05233595624294383 sin4=0.0697564737441253sin5=0.08715574274765816sin6=0.10452846326765346 sin7=0.12186934340514747sin8=0.13917310096006544sin9=0.15643446504023087 sin10=0.17364817766693033sin11=0.1908089953765448sin12=0.20791169081775931 sin13=0.22495105434386497sin14=0.24192189559966773sin15=0.25881904510252074 sin16=0.27563735581699916sin17=0.2923717047227367sin18=0.3090169943749474 sin19=0.3255681544571567sin20=0.3420201433256687sin21=0.35836794954530027 sin22=0.374606593415912sin23=0.3907311284892737sin24=0.40673664307580015 sin25=0.42261826174069944sin26=0.4383711467890774sin27=0.45399049973954675 sin28=0.4694715627858908sin29=0.48480962024633706sin30=0.49999999999999994 sin31=0.5150380749100542sin32=0.5299192642332049sin33=0.544639035015027 sin34=0.5591929034707468sin35=0.573576436351046sin36=0.5877852522924731 sin37=0.6018150231520483sin38=0.6156614753256583sin39=0.6293203910498375 sin40=0.6427876096865392sin41=0.6560590289905073sin42=0.6691306063588582 sin43=0.6819983600624985sin44=0.6946583704589972sin45=0.7071067811865475 sin46=0.7193398003386511sin47=0.7313537016191705sin48=0.7431448254773941 sin49=0.7547095802227719sin50=0.766044443118978sin51=0.7771459614569708 sin52=0.7880107536067219sin53=0.7986355100472928sin54=0.8090169943749474 sin55=0.8191520442889918sin56=0.8290375725550417sin57=0.8386705679454239 sin58=0.848048096156426sin59=0.8571673007021122sin60=0.8660254037844386 sin61=0.8746197071393957sin62=0.8829475928589269sin63=0.8910065241883678 sin64=0.898794046299167sin65=0.9063077870366499sin66=0.9135454576426009 sin67=0.9205048534524404sin68=0.9271838545667873sin69=0.9335804264972017 sin70=0.9396926207859083sin71=0.9455185755993167sin72=0.9510565162951535 sin73=0.9563047559630354sin74=0.9612616959383189sin75=0.9659258262890683 sin76=0.9702957262759965sin77=0.9743700647852352sin78=0.9781476007338057 sin79=0.981627183447664sin80=0.984807753012208sin81=0.9876883405951378 sin82=0.9902680687415704sin83=0.992546151641322sin84=0.9945218953682733 sin85=0.9961946980917455sin86=0.9975640502598242sin87=0.9986295347545738 sin88=0.9993908270190958sin89=0.9998476951563913 sin90=1 cos1=0.9998476951563913cos2=0.9993908270190958cos3=0.9986295347545738 cos4=0.9975640502598242cos5=0.9961946980917455cos6=0.9945218953682733 cos7=0.992546151641322cos8=0.9902680687415704cos9=0.9876883405951378 cos10=0.984807753012208cos11=0.981627183447664cos12=0.9781476007338057 cos13=0.9743700647852352cos14=0.9702957262759965cos15=0.9659258262890683 cos16=0.9612616959383189cos17=0.9563047559630355cos18=0.9510565162951535 cos19=0.9455185755993168cos20=0.9396926207859084cos21=0.9335804264972017 cos22=0.9271838545667874cos23=0.9205048534524404cos24=0.9135454576426009 cos25=0.9063077870366499cos26=0.898794046299167cos27=0.8910065241883679 cos28=0.882947592858927cos29=0.8746197071393957cos30=0.8660254037844387 cos31=0.8571673007021123cos32=0.848048096156426cos33=0.838670567945424 cos34=0.8290375725550417cos35=0.8191520442889918cos36=0.8090169943749474 cos37=0.7986355100472928cos38=0.7880107536067219cos39=0.7771459614569709 cos40=0.766044443118978cos41=0.754709580222772cos42=0.7431448254773942 cos43=0.7313537016191705cos44=0.7193398003386512cos45=0.7071067811865476 cos46=0.6946583704589974cos47=0.6819983600624985cos48=0.6691306063588582 cos49=0.6560590289905074cos50=0.6427876096865394cos51=0.6293203910498375 cos52=0.6156614753256583cos53=0.6018150231520484cos54=0.5877852522924731 cos55=0.5735764363510462cos56=0.5591929034707468cos57=0.5446390350150272 cos58=0.5299192642332049cos59=0.5150380749100544cos60=0.5000000000000001 cos61=0.4848096202463371cos62=0.46947156278589086cos63=0.4539904997395468 cos64=0.43837114678907746cos65=0.42261826174069944cos66=0.4067366430758004 cos67=0.3907311284892737cos68=0.3746065934159122cos69=0.35836794954530015 cos70=0.3420201433256688cos71=0.32556815445715675cos72=0.30901699437494745 cos73=0.29237170472273677cos74=0.27563735581699916cos75=0.25881904510252074 cos76=0.24192189559966767cos77=0.22495105434386514cos78=0.20791169081775923 cos79=0.19080899537654491cos80=0.17364817766693041cos81=0.15643446504023092 cos82=0.13917310096006546cos83=0.12186934340514749cos84=0.10452846326765346 cos85=0.08715574274765836cos86=0.06975647374412523cos87=0.052335956242943966 cos88=0.03489949670250108cos89=0.0174524064372836 cos90=0 tan1=0.017455064928217585tan2=0.03492076949174773tan3=0.052407779283041196 tan4=0.06992681194351041tan5=0.08748866352592401tan6=0.10510423526567646 tan7=0.1227845609029046tan8=0.14054083470239145tan9=0.15838444032453627 tan10=0.17632698070846497tan11=0.19438030913771848tan12=0.2125565616700221 tan13=0.2308681911255631tan14=0.24932800284318068tan15=0.2679491924311227 tan16=0.2867453857588079tan17=0.30573068145866033tan18=0.3249196962329063 tan19=0.34432761328966527tan20=0.36397023426620234tan21=0.3838640350354158 tan22=0.4040262258351568tan23=0.4244748162096047tan24=0.4452286853085361 tan25=0.4663076581549986tan26=0.4877325885658614tan27=0.5095254494944288 tan28=0.5317094316614788tan29=0.554309051452769tan30=0.5773502691896257 tan31=0.6008606190275604tan32=0.6248693519093275tan33=0.6494075931975104 tan34=0.6745085168424265tan35=0.7002075382097097tan36=0.7265425280053609 tan37=0.7535540501027942tan38=0.7812856265067174tan39=0.8097840331950072 tan40=0.8390996311772799tan41=0.8692867378162267tan42=0.9004040442978399 tan43=0.9325150861376618tan44=0.9656887748070739tan45=0.9999999999999999 tan46=1.0355303137905693tan47=1.0723687100246826tan48=1.1106125148291927 tan49=1.1503684072210092tan50=1.19175359259421tan51=1.234897156535051 tan52=1.2799416321930785tan53=1.3270448216204098tan54=1.3763819204711733 tan55=1.4281480067421144tan56=1.4825609685127403tan57=1.5398649638145827 tan58=1.6003345290410506tan59=1.6642794823505173tan60=1.7320508075688767 tan61=1.8040477552714235tan62=1.8807264653463318tan63=1.9626105055051503 tan64=2.050303841579296tan65=2.1445069205095586tan66=2.246036773904215 tan67=2.355852365823753tan68=2.4750868534162946tan69=2.6050890646938023 tan70=2.7474774194546216tan71=2.904210877675822tan72=3.0776835371752526 tan73=3.2708526184841404tan74=3.4874144438409087tan75=3.7320508075688776 tan76=4.0107809335358455tan77=4.331475874284153tan78=4.704630109478456 tan79=5.144554015970307tan80=5.671281819617707tan81=6.313751514675041 tan82=7.115369722384207tan83=8.144346427974593tan84=9.514364454222587 tan85=11.43005230276132tan86=14.300666256711942tan87=19.08113668772816 tan88=28.636253282915515tan89=57.289961630759144 tan90=无取值HYPERLINK"http://www.baike.com/wiki/%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%E5%80%BC"\l"#"\t"_self"三角函数值-数学方程式数关系 tanα·cotα=1 sinα·cscα=1 cosα·secα=1商数关系tanα=sinα/cosαcotα=cosα/sinα平方关系 sinα²+cosα²=1 1+tanα²=secα² 1+cotα²=cscα² 以下关系,函数名不变,图像看象限. sin(2kπ+α)=sinα cos(2kπ+α)=cosα tan(2kπ+α)=tanα cot(2kπ+α)=cotα sin(π+α)=-sinα cos(π+α)=-cosα tan(π+α)=tanα cot(π+α)=cotα sin(π-α)=sinα cos(π-α)=-cosα tan(π-α)=-tanα cot(π-α)=-cotα sin(2π-α)=-sinα cos(2π-α)=cosα tan(2π-α)=-tanα cot(2π-α)=-cotα 以下关系,奇变偶不变,符号看象限 sin(90°-α)=cosα cos(90°-α)=sinα tan(90°-α)=cotα cot(90°-α)=tanα sin(90°+α)=cosα cos(90°+α)=-sinα tan(90°+α)=-cotα cot(90°+α)=-tanα sin(270°-α)=-cosα cos(270°-α)=-sinα tan(270°-α)=cotα cot(270°-α)=tanα sin(270°+α)=-cosα cos(270°+α)=sinα tan(270°+α)=-cotα cot(270°+α)=-tanα积化合差公式 sinα·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)]和差化积公式 sinα+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]三倍角公式 sin3α=3sinα-4sinα³ cos3α=4cosα³-3cosα 两角和与差的三角函数关系 sin(α+β)=sinαcosβ+cosαsinβ sin(α-β)=sinαcosβ-cosαsinβ cos(α+β)=cosαcosβ-sinαsinβ cos(α-β)=cosαcosβ+sinαsinβ tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ) tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)正弦二倍角公式: sin2α=2cosαsinα 推导:sin2A=sin(A+A)=sinAcosA+cosAsinA=2sinAcosA 拓展公式:sin2A=2sinAcosA=2tanAcos^2A=2tanA/[1+tan^2A] 1+sin2A=(sinA+cosA)^2余弦二倍角公式: 余弦二倍角公式有三组表示形式,三组形式等价: 1.Cos2a=Cos^2a-Sin^2a=[1-tan^2a]/[1+tan^2a] 2.Cos2a=1-2Sin^2a 3.Cos2a=2Cos^2a-1 推导:cos2A=cos(A+A)=cosAcosA-sinAsinA=cos^2A-sin^2A=2cos^2A-1 =1-2sin^2A正切二倍角公式: tan2α=2tanα/[1-tan^2α] 推导:tan2A=tan(A+A)=(tanA+tanA)/(1-tanAtanA)=2tanA/[1-tan^2A]降幂公式: cosA^2=[1+cos2A]/2 sinA^2=[1-cos2A]/2 tanA^2=[1-cos2A]/[1+cos2A] 变式: sin2α=sin^2(α+π/4)-cos^2(α+π/4)=2sin^2(a+π/4)-1=1-2cos^2(α+π/4); cos2α=2sin(α+π/4)cos(α+π/4) 余弦定理: a^2=b^2+c^2-2bccosA b^2=c^2+a^2-2cacosB c^2=a^2+b^2-2abcosC