數值表
| 角α | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° | 270° | 360° |
| 弧度制 | o | π/6 | π/4 | π/3 | π/2 | 2π/3 | 3π/4 | 5π/6 | π | 3π/2 | 2π |
| sinα | o | 1/2 | √2/2 | √3/2 | 1 | √3/2 | √2/2 | 1/2 | 0 | -1 | 0 |
| cosα | 1 | √3/2 | √2/2 | 1/2 | 0 | -1/2 | -√2/2 | -√3/2 | -1 | 0 | 1 |
| tanα | o | √3/3 | 1 | √3 | ±∞ | -√3 | -1 | -√3/3 | 0 | ±∞ | 0 |
三角函式值cos0=cos0°=1
tan0=tan0°=0
sin15=0.650；sin15°=（√6-√2）/4
cos15=-0.759；cos15°=（√6+√2）/4
tan15=-0.855；tan15°=2-√3
sin30=-0.988；sin30°=1/2
cos30=0.154；cos30°=√3/2
tan30=-6.405；tan30°=√3/3
sin45=0.851；sin45°=√2/2
cos45=0.525；cos45°=sin45°=√2/2
tan45=1.620；tan45°=1
sin60=-0.305；sin60°=√3/2
cos60=-0.952；cos60°=1/2
tan60=0.320；tan60°=√3
sin75=-0.388；sin75°=cos15°
cos75=0.922；cos75°=sin15°
tan75=-0.421；tan75°=sin75°/cos75°=2+√3
sin90=0.894；sin90°=cos0°=1
cos90=-0.448；cos90°=sin0°=0
tan90=-1.995；tan90°不存在
sin105=-0.971；sin105°=cos15°
cos105=-0.241；cos105°=-sin15°
tan105=4.028；tan105°=-cot15°
sin120=0.581；sin120°=cos30°
cos120=0.814；cos120°=-sin30°
tan120=0.713；tan120°=-tan60°
sin135=0.088；sin135°=sin45°
cos135=-0.996；cos135°=-cos45°
tan135=-0.0887；tan135°=-tan45°
sin150=-0.7149；sin150°=sin30°
cos150=-0.699；cos150°=-cos30°
tan150=-1.022；tan150°=-tan30°
sin165=0.998；sin165°=sin15°
cos165=-0.066；cos165°=-cos15°
tan165=-15.041；tan165°=-tan15°
sin180=-0.801；sin180°=sin0°=0
cos180=-0.598；cos180°=-cos0°=-1
tan180=1.339；tan180°=0
sin195=0.219；sin195°=-sin15°
cos195=0.976；cos195°=-cos15°
tan195=0.225；tan195°=tan15°
sin360=0.959；sin360°=sin0°=0
cos360=-0.284；cos360°=cos0°=1
tan360=-3.380；tan360°=tan0°=0
cos72=[(√5)-1]/4（利用黃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=無取值
數學方程式
數關係
tanα·cotα=1
sinα·cscα=1
cosα·secα=1
商的關係
tanα=sinα/cosαcotα=cosα/sinα
平方關係
三角函式值1+tanα²=secα²
1+cotα²=cscα²
以下關係，函式名不變，符號看象限.
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
以下關係，奇變偶不變，符號看象限
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
積化合差公式
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）]
和差化積公式
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
兩角和與差的三角函式關係
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
推導：sin2A=sin(A+A)=sinAcosA+cosAsinA=2sinAcosA
拓展公式：sin2A=2sinAcosA=2tanAcos2A=2tanA/[1+tan2A]
1+sin2A=(sinA+cosA)^2
餘弦二倍角公式
餘弦二倍角公式有三組表示形式，三組形式等價：
1.Cos2a=Cos2a-Sin2a=[1-tan2a]/[1+tan2a]
2.Cos2a=1-2Sin2a
3.Cos2a=2Cos2a-1
推導：cos2A=cos(A+A)=cosAcosA-sinAsinA=cos2A-sin2A=2cos2A-1
=1-2sin2A
正切二倍角公式
tan2α=2tanα/[1-tan2α]
推導：tan2A=tan(A+A)=(tanA+tanA)/(1-tanAtanA)=2tanA/[1-tan2A]
降冪公式
cosA^2=[1+cos2A]/2
sinA^2=[1-cos2A]/2
tanA^2=[1-cos2A]/[1+cos2A]
變式
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)
餘弦定理
a^2=b^2+c²-2bccosA
b^2=c^2+a^2-2cacosB
c^2=a^2+b^2-2abcosC
套用
三角函式值它有六種基本函式：
函式名正弦餘弦正切餘切正割餘割
符號sincostancotseccsc
正弦函式sin（A）=a/c
餘弦函式cos（A）=b/c
正切函式tan（A）=a/b
餘切函式cot（A）=b/a
其中a為對邊，b為鄰邊，c為斜邊
