初中數(shù)學(xué)考點(diǎn)之三角形3篇

第初中數(shù)學(xué)考點(diǎn)之三角形3篇

初中數(shù)學(xué)知識(shí)點(diǎn)：銳角三角函數(shù)的定義

銳角三角函數(shù)：

銳角角A的正弦(sin),余弦(cos)和正切(tan),余切(cot)以及正割(sec)，余割(csc)都叫做角A的銳角三角函數(shù)。

初中學(xué)習(xí)的銳角三角函數(shù)值的定義方法是在直角三角形中定義的，所以在初中階段求銳角的三角函數(shù)值，都是通過(guò)構(gòu)造直角三角形來(lái)完成的，即把這個(gè)角放到某個(gè)直角三角形中。所謂銳角三角函數(shù)是指：我們初中研究的都是銳角的三角函數(shù)。初中研究的銳角的三角函數(shù)為：正弦(sin)，余弦(cos)，正切(tan)。

銳角三角函數(shù)的增減性：當(dāng)角度在0°~90°之間變化時(shí)：

(1)正弦值隨著角度的增大而增大;

(2)余弦值隨著角度的增大而減小;

(3)正切值隨著角度的增大而增大。

銳角三角函數(shù)的增減性：

1.銳角三角函數(shù)值都是正值

2.當(dāng)角度在0°～90°間變化時(shí)，

正弦值隨著角度的增大(或減小)而增大(或減小)，余弦值隨著角度的增大(或減小)而減小(或增大);

正切值隨著角度的增大(或減小)而增大(或減小)，余切值隨著角度的增大(或減小)而減小(或增大);

正割值隨著角度的增大(或減小)而增大(或減小)，余割值隨著角度的增大(或減小)而減小(或增大)。

3.當(dāng)角度在0°≤A≤90°間變化時(shí)，0≤sinA≤1,1≥cosA≥0;當(dāng)角度在0°0。

銳角三角函數(shù)的關(guān)系式：

同角三角函數(shù)基本關(guān)系式

tanα·cotα=1

sin2α·cos2α=1

cos2α·sin2α=1

sinα/cosα=tanα=secα/cscα

cosα/sinα=cotα=cscα/secα

(sinα)2+(cosα)2=1

1+tanα=secα

1+cotα=cscα

誘導(dǎo)公式

sin(-α)=-sinα

cos(-α)=cosα

tan(-α)=-tanα

cot(-α)=-cotα

sin(π/2-α)=cosα

cos(π/2-α)=sinα

tan(π/2-α)=cotα

cot(π/2-α)=tanα

sin(π/2+α)=cosα

cos(π/2+α)=-sinα

tan(π/2+α)=-cotα

cot(π/2+α)=-tanα

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(3π/2-α)=-cosα

cos(3π/2-α)=-sinα

tan(3π/2-α)=cotα

cot(3π/2-α)=tanα

sin(3π/2+α)=-cosα

cos(3π/2+α)=sinα

tan(3π/2+α)=-cotα

cot(3π/2+α)=-tanα

sin(2π-α)=-sinα

cos(2π-α)=cosα

tan(2π-α)=-tanα

cot(2π-α)=-cotα

sin(2kπ+α)=sinα

cos(2kπ+α)=cosα

tan(2kπ+α)=tanα

cot(2kπ+α)=cotα(其中k∈Z)

二倍角、三倍角的正弦、余弦和正切公式

Sin(2α)=2sinαcosα

Cos(2α)=(cosα)2-(sinα)2=2(cosα)2-1=1-2(sinα)2

Tan(2α)=2tanα/(1-tanα)

sin(3α)=3sinα-4sin3α=4sinα·sin(60°+α)sin(60°-α)

cos(3α)=4cos3α-3cosα=4cosα·cos(60°+α)cos(60°-α)

tan(3α)=(3tanα-tan3α)/(1-3tan2α)=tanαtan(π/3+α)tan(π/3-α)

和差化積、積化和差公式

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]

sinαcosβ=-[sin(α+β)+sin(α-β)]

sinαsinβ=-[1][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

初中數(shù)學(xué)知識(shí)點(diǎn)：同角三角函數(shù)的關(guān)系

三種基本題型:

①三角函數(shù)值的計(jì)算問(wèn)題：利用平方關(guān)系時(shí)，往往要開(kāi)方，因此要先根據(jù)角的所在象限確定符號(hào)，即將角所在象限進(jìn)行分類討論。

②化簡(jiǎn)題：一定要在有意義的前提下進(jìn)行。

③證明問(wèn)題。

三類：

同角三角函數(shù)的基本關(guān)系：

(sinθ)2+(cosθ)2=1;

tanθcotθ=sinθcscθ=cosθsecθ=1;

(secθ)2-(tanθ)2=(cscθ)2-(cosθ)2=1

誘導(dǎo)公式，在360°內(nèi)的變換(角度制)：

取值sinθcosθtanθ

αsinαcosαtanα

-α-sinαcosα-tanα

180+α-sinα-cosαtanα

180-αsinα-cosα-tanα

360+αsinαcosαtanα

360-α-sinαcosα-tanα

90+αcosα-sinα-cotα

90-αcosαsinαcotα

270+α-cosαsinα-cotα

270-α-cosα-sinαcotα

兩個(gè)角的變換關(guān)系，不屬于初中內(nèi)容：

sin(α+β)=sinαcosβ+cosαsinβ

sin(α-β)=sinαcosβ-cosαsinβ

cos(α+β)=cosαcosβ-sinαsinβ

cos(α-β)=cosαcosβ+sinαsinβ

以此四個(gè)公式為基礎(chǔ)，可推導(dǎo)出其他公式。

初中數(shù)學(xué)知識(shí)點(diǎn)：解直角三角形

概念：

在直角三角形中，除直角外，一共有五個(gè)元素，即三條邊和兩個(gè)銳角，由直角三角形中除直角外的已知元素，求出所有未知元素的過(guò)程叫做解直角三角形。

解直角三角形的邊角關(guān)系：

在Rt△ABC中，∠C=90°，∠A，∠B，∠C所對(duì)的邊分別為a，b，c，

解直角三角形的應(yīng)用：

一般步驟是：

(1)將實(shí)際問(wèn)題抽象為數(shù)學(xué)問(wèn)題(畫(huà)圖，轉(zhuǎn)化為直角三角形的問(wèn)題);

(2)根據(jù)題目的條件，適當(dāng)選擇銳角三角函數(shù)等去解三角形;

(3)得到數(shù)學(xué)問(wèn)題的答案;

(4)還原為實(shí)際問(wèn)題的答案。

銳角三角函數(shù)：

sinA=a/c,cosA=b/c,tanA=a/b,cotA=b/a

(1)互余角的三角函數(shù)值之間的關(guān)系：

若∠A+∠B=90°，那么sinA=cosB或sinB=cosA

(2)同角的三角函數(shù)值之間的關(guān)系：

①sin2A+cos2A=1

②tanA=sinA/cosA

③tanA=1/tanB

④a/sinA=b/sinB=c/sinC

(3)銳角三角函數(shù)隨角度的變化規(guī)律：

銳角∠A的tan值和sin值隨著角度的增大而增大，cos值隨著角度的增大而減小。

解直角三角形的函數(shù)值列舉：

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=0.3420232333256687sin21=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=0.9335804264972023

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=0.9335804264972023

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=0.3420232333256688cos71=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=1.1917535925