三角函数值表知识汇总

对于三角函数公式,学生容易记错。为使学生记住并能熟练地应用,教师还要帮助学生掌握记忆这些公式的方法。本文是小编整理三角函数值表知识汇总的.资料，仅供参考。

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=0.01745240643728351 sin2=0.03489949670250097 sin3=0.05233595624294383

sin4=0.0697564737441253 sin5=0.08715574274765816 sin6=0.10452846326765346

sin7=0.12186934340514747 sin8=0.13917310096006544 sin9=0.15643446504023087

sin10=0.17364817766693033 sin11=0.1908089953765448 sin12=0.20791169081775931

sin13=0.22495105434386497 sin14=0.24192189559966773 sin15=0.25881904510252074

sin16=0.27563735581699916 sin17=0.2923717047227367 sin18=0.3090169943749474

sin19=0.3255681544571567 sin20=0.3420201433256687 sin21=0.35836794954530027

sin22=0.374606593415912 sin23=0.3907311284892737 sin24=0.40673664307580015

sin25=0.42261826174069944 sin26=0.4383711467890774 sin27=0.45399049973954675

sin28=0.4694715627858908 sin29=0.48480962024633706 sin30=0.49999999999999994

sin31=0.5150380749100542 sin32=0.5299192642332049 sin33=0.544639035015027

sin34=0.5591929034707468 sin35=0.573576436351046 sin36=0.5877852522924731

sin37=0.6018150231520483 sin38=0.6156614753256583 sin39=0.6293203910498375

sin40=0.6427876096865392 sin41=0.6560590289905073 sin42=0.6691306063588582

sin43=0.6819983600624985 sin44=0.6946583704589972 sin45=0.7071067811865475

sin46=0.7193398003386511 sin47=0.7313537016191705 sin48=0.7431448254773941

sin49=0.7547095802227719 sin50=0.766044443118978 sin51=0.7771459614569708

sin52=0.7880107536067219 sin53=0.7986355100472928 sin54=0.8090169943749474

sin55=0.8191520442889918 sin56=0.8290375725550417 sin57=0.8386705679454239

sin58=0.848048096156426 sin59=0.8571673007021122 sin60=0.8660254037844386

sin61=0.8746197071393957 sin62=0.8829475928589269 sin63=0.8910065241883678

sin64=0.898794046299167 sin65=0.9063077870366499 sin66=0.9135454576426009

sin67=0.9205048534524404 sin68=0.9271838545667873 sin69=0.9335804264972017

sin70=0.9396926207859083 sin71=0.9455185755993167 sin72=0.9510565162951535

sin73=0.9563047559630354 sin74=0.9612616959383189 sin75=0.9659258262890683

sin76=0.9702957262759965 sin77=0.9743700647852352 sin78=0.9781476007338057

sin79=0.981627183447664 sin80=0.984807753012208 sin81=0.9876883405951378

sin82=0.9902680687415704 sin83=0.992546151641322 sin84=0.9945218953682733

sin85=0.9961946980917455 sin86=0.9975640502598242 sin87=0.9986295347545738

sin88=0.9993908270190958 sin89=0.9998476951563913

sin90=1

cos1=0.9998476951563913 cos2=0.9993908270190958 cos3=0.9986295347545738

cos4=0.9975640502598242 cos5=0.9961946980917455 cos6=0.9945218953682733

cos7=0.992546151641322 cos8=0.9902680687415704 cos9=0.9876883405951378

cos10=0.984807753012208 cos11=0.981627183447664 cos12=0.9781476007338057

cos13=0.9743700647852352 cos14=0.9702957262759965 cos15=0.9659258262890683

cos16=0.9612616959383189 cos17=0.9563047559630355 cos18=0.9510565162951535

cos19=0.9455185755993168 cos20=0.9396926207859084 cos21=0.9335804264972017

cos22=0.9271838545667874 cos23=0.9205048534524404 cos24=0.9135454576426009

cos25=0.9063077870366499 cos26=0.898794046299167 cos27=0.8910065241883679

cos28=0.882947592858927 cos29=0.8746197071393957 cos30=0.8660254037844387

cos31=0.8571673007021123 cos32=0.848048096156426 cos33=0.838670567945424

cos34=0.8290375725550417 cos35=0.8191520442889918 cos36=0.8090169943749474

cos37=0.7986355100472928 cos38=0.7880107536067219 cos39=0.7771459614569709

cos40=0.766044443118978 cos41=0.754709580222772 cos42=0.7431448254773942

cos43=0.7313537016191705 cos44=0.7193398003386512 cos45=0.7071067811865476

cos46=0.6946583704589974 cos47=0.6819983600624985 cos48=0.6691306063588582

cos49=0.6560590289905074 cos50=0.6427876096865394 cos51=0.6293203910498375

cos52=0.6156614753256583 cos53=0.6018150231520484 cos54=0.5877852522924731

cos55=0.5735764363510462 cos56=0.5591929034707468 cos57=0.5446390350150272

cos58=0.5299192642332049 cos59=0.5150380749100544 cos60=0.5000000000000001

cos61=0.4848096202463371 cos62=0.46947156278589086 cos63=0.4539904997395468

cos64=0.43837114678907746 cos65=0.42261826174069944 cos66=0.4067366430758004

cos67=0.3907311284892737 cos68=0.3746065934159122 cos69=0.35836794954530015

cos70=0.3420201433256688 cos71=0.32556815445715675 cos72=0.30901699437494745

cos73=0.29237170472273677 cos74=0.27563735581699916 cos75=0.25881904510252074

cos76=0.24192189559966767 cos77=0.22495105434386514 cos78=0.20791169081775923

cos79=0.19080899537654491 cos80=0.17364817766693041 cos81=0.15643446504023092

cos82=0.13917310096006546 cos83=0.12186934340514749 cos84=0.10452846326765346

cos85=0.08715574274765836 cos86=0.06975647374412523 cos87=0.052335956242943966

cos88=0.03489949670250108 cos89=0.0174524064372836

cos90=0

tan1=0.017455064928217585 tan2=0.03492076949174773 tan3=0.052407779283041196

tan4=0.06992681194351041 tan5=0.08748866352592401 tan6=0.10510423526567646

tan7=0.1227845609029046 tan8=0.14054083470239145 tan9=0.15838444032453627

tan10=0.17632698070846497 tan11=0.19438030913771848 tan12=0.2125565616700221

tan13=0.2308681911255631 tan14=0.24932800284318068 tan15=0.2679491924311227

tan16=0.2867453857588079 tan17=0.30573068145866033 tan18=0.3249196962329063

tan19=0.34432761328966527 tan20=0.36397023426620234 tan21=0.3838640350354158

tan22=0.4040262258351568 tan23=0.4244748162096047 tan24=0.4452286853085361

tan25=0.4663076581549986 tan26=0.4877325885658614 tan27=0.5095254494944288

tan28=0.5317094316614788 tan29=0.554309051452769 tan30=0.5773502691896257

tan31=0.6008606190275604 tan32=0.6248693519093275 tan33=0.6494075931975104

tan34=0.6745085168424265 tan35=0.7002075382097097 tan36=0.7265425280053609

tan37=0.7535540501027942 tan38=0.7812856265067174 tan39=0.8097840331950072

tan40=0.8390996311772799 tan41=0.8692867378162267 tan42=0.9004040442978399

tan43=0.9325150861376618 tan44=0.9656887748070739 tan45=0.9999999999999999

tan46=1.0355303137905693 tan47=1.0723687100246826 tan48=1.1106125148291927

tan49=1.1503684072210092 tan50=1.19175359259421 tan51=1.234897156535051

tan52=1.2799416321930785 tan53=1.3270448216204098 tan54=1.3763819204711733

tan55=1.4281480067421144 tan56=1.4825609685127403 tan57=1.5398649638145827

tan58=1.6003345290410506 tan59=1.6642794823505173 tan60=1.7320508075688767

tan61=1.8040477552714235 tan62=1.8807264653463318 tan63=1.9626105055051503

tan64=2.050303841579296 tan65=2.1445069205095586 tan66=2.246036773904215

tan67=2.355852365823753 tan68=2.4750868534162946 tan69=2.6050890646938023

tan70=2.7474774194546216 tan71=2.904210877675822 tan72=3.0776835371752526

tan73=3.2708526184841404 tan74=3.4874144438409087 tan75=3.7320508075688776

tan76=4.0107809335358455 tan77=4.331475874284153 tan78=4.704630109478456

tan79=5.144554015970307 tan80=5.671281819617707 tan81=6.313751514675041

tan82=7.115369722384207 tan83=8.144346427974593 tan84=9.514364454222587

tan85=11.43005230276132 tan86=14.300666256711942 tan87=19.08113668772816

tan88=28.636253282915515 tan89=57.289961630759144

tan90=无取值

锐角三角函数公式

sin α=∠α的对边 / 斜边

cos α=∠α的邻边 / 斜边

tan α=∠α的对边 / ∠α的邻边

cot α=∠α的邻边 / ∠α的对边

倍角公式

Sin2A=2SinA?CosA

Cos2A=CosA^2-SinA^2=1-2SinA^2=2CosA^2-1

tan2A=(2tanA)/(1-tanA^2)

(注:SinA^2 是sinA的平方 sin2(A) )

三倍角公式

sin3α=4sinα·sin(π/3+α)sin(π/3-α)

cos3α=4cosα·cos(π/3+α)cos(π/3-α)

tan3a = tan a · tan(π/3+a)· tan(π/3-a)

三倍角公式推导

sin3a

=sin(2a+a)

=sin2acosa+cos2asina

辅助角公式

Asinα+Bcosα=(A^2+B^2)^(1/2)sin(α+t)，其中

sint=B/(A^2+B^2)^(1/2)

cost=A/(A^2+B^2)^(1/2)

tant=B/A

Asinα+Bcosα=(A^2+B^2)^(1/2)cos(α-t)，tant=A/B

降幂公式

sin^2(α)=(1-cos(2α))/2=versin(2α)/2

cos^2(α)=(1+cos(2α))/2=covers(2α)/2

tan^2(α)=(1-cos(2α))/(1+cos(2α))

推导公式

tanα+cotα=2/sin2α

tanα-cotα=-2cot2α

1+cos2α=2cos^2α

1-cos2α=2sin^2α

1+sinα=(sinα/2+cosα/2)^2

=2sina(1-sin²a)+(1-2sin²a)sina

=3sina-4sin³a

cos3a

=cos(2a+a)

=cos2acosa-sin2asina

=(2cos²a-1)cosa-2(1-sin²a)cosa

=4cos³a-3cosa

sin3a=3sina-4sin³a

=4sina(3/4-sin²a)

=4sina[(√3/2)²-sin²a]

=4sina(sin²60°-sin²a)

=4sina(sin60°+sina)(sin60°-sina)

=4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°-a)/2]

=4sinasin(60°+a)sin(60°-a)

cos3a=4cos³a-3cosa

=4cosa(cos²a-3/4)

=4cosa[cos²a-(√3/2)²]

=4cosa(cos²a-cos²30°)

=4cosa(cosa+cos30°)(cosa-cos30°)

=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]}

=-4cosasin(a+30°)sin(a-30°)

=-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]

=-4cosacos(60°-a)[-cos(60°+a)]

=4cosacos(60°-a)cos(60°+a)

上述两式相比可得

tan3a=tanatan(60°-a)tan(60°+a)

半角公式

tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA);

cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA.

sin^2(a/2)=(1-cos(a))/2

cos^2(a/2)=(1+cos(a))/2

tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))

学习方法网[]

三角和

sin(α+β+γ)=sinα·cosβ·cosγ+cosα·sinβ·cosγ+cosα·cosβ·sinγ-sinα·sinβ·sinγ

cos(α+β+γ)=cosα·cosβ·cosγ-cosα·sinβ·sinγ-sinα·cosβ·sinγ-sinα·sinβ·cosγ

tan(α+β+γ)=(tanα+tanβ+tanγ-tanα·tanβ·tanγ)/(1-tanα·tanβ-tanβ·tanγ-tanγ·tanα)

两角和差

cos(α+β)=cosα·cosβ-sinα·sinβ

cos(α-β)=cosα·cosβ+sinα·sinβ

sin(α±β)=sinα·cosβ±cosα·sinβ

tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)

tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)

和差化积

sinθ+sinφ = 2 sin[(θ+φ)/2] cos[(θ-φ)/2]

sinθ-sinφ = 2 cos[(θ+φ)/2] sin[(θ-φ)/2]

cosθ+cosφ = 2 cos[(θ+φ)/2] cos[(θ-φ)/2]

cosθ-cosφ = -2 sin[(θ+φ)/2] sin[(θ-φ)/2]

tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB)

tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)

积化和差

sinαsinβ = [cos(α-β)-cos(α+β)] /2

cosαcosβ = [cos(α+β)+cos(α-β)]/2

sinαcosβ = [sin(α+β)+sin(α-β)]/2

cosαsinβ = [sin(α+β)-sin(α-β)]/2

诱导公式

sin(-α) = -sinα

cos(-α) = cosα

tan (—a)=-tanα

sin(π/2-α) = cosα

cos(π/2-α) = sinα

sin(π/2+α) = cosα

cos(π/2+α) = -sinα

sin(π-α) = sinα

cos(π-α) = -cosα

sin(π+α) = -sinα

cos(π+α) = -cosα

tanA= sinA/cosA

tan(π/2+α)=-cotα

tan(π/2-α)=cotα

tan(π-α)=-tanα

tan(π+α)=tanα

诱导公式记背诀窍:奇变偶不变，符号看象限

万能公式

sinα=2tan(α/2)/[1+tan^(α/2)]

cosα=[1-tan^(α/2)]/1+tan^(α/2)]

tanα=2tan(α/2)/[1-tan^(α/2)]

其它公式

(1)(sinα)^2+(cosα)^2=1

(2)1+(tanα)^2=(secα)^2

(3)1+(cotα)^2=(cscα)^2

证明下面两式,只需将一式,左右同除(sinα)^2,第二个除(cosα)^2即可

(4)对于任意非直角三角形,总有

tanA+tanB+tanC=tanAtanBtanC

证:

A+B=π-C

tan(A+B)=tan(π-C)

(tanA+tanB)/(1-tanAtanB)=(tanπ-tanC)/(1+tanπtanC)

整理可得

tanA+tanB+tanC=tanAtanBtanC

得证

同样可以得证,当x+y+z=nπ(n∈Z)时,该关系式也成立

由tanA+tanB+tanC=tanAtanBtanC可得出以下结论

(5)cotAcotB+cotAcotC+cotBcotC=1

(6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2)

(7)(cosA)^2+(cosB)^2+(cosC)^2=1-2cosAcosBcosC

(8)(sinA)^2+(sinB)^2+(sinC)^2=2+2cosAcosBcosC

(9)sinα+sin(α+2π/n)+sin(α+2π*2/n)+sin(α+2π*3/n)+……+sin[α+2π*(n-1)/n]=0

cosα+cos(α+2π/n)+cos(α+2π*2/n)+cos(α+2π*3/n)+……+cos[α+2π*(n-1)/n]=0 以及

sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2

tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0