三角函數(shù)關(guān)系式大全

同角三角函數(shù)關(guān)系式三三角函數(shù)sin^2(α)+cos^2(α)=1cos^2(a)=1-sin^2(a)tan^2(α)+1=1/cos^2(α)2sin^2(a)=1-cos2(a)cot^2(α)+1=1/sin^2(a)sinα=tanα×cosαcosα=cotα×sinαtanα=sinα×secαcotα=cosα×cscαsecα=tanα×cscαcscα=secα×cotα13實用精品文檔sinα/cosα＝tanα＝secα/cscαcosα/sinα＝cotα＝cscα/secαABCA角A的對邊比斜邊,A正切等于對邊比鄰邊,設(shè)α為任意角，終邊相同的角的同一三角函數(shù)的值相等：精品文檔三三角函數(shù)sin(2kπ＋α)＝sinαcos(2kπ＋α)＝cosαtan(2kπ＋α)＝tanαcot(2kπ＋α)＝cotαsin(π＋α)＝－sinαcos(π＋α)＝－cosαtan(π＋α)＝tanαcot(π＋α)＝cotαsin(－α)＝－sinαcos(－α)＝cosαtan(－α)＝－tanαcot(－α)＝－cotαsin(π－α)＝sinαcos(π－α)＝－cosα/13實用精品文檔tan(π－α)＝－tanαcot(π－α)＝－cotαsin(2π－α)＝－sinαcos(2π－α)＝cosαtan(2π－α)＝－tanαcot(2π－α)＝－cotαsin(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(3π/2＋α)＝－cosαcos(3π/2＋α)＝sinαtan(3π/2＋α)＝－cotα3實用精品文檔cot(3π/2＋α)＝－tanαsin(3π/2－α)＝－cosαcos(3π/2－α)＝－sinαtan(3π/2－α)＝cotαcot(3π/2－α)＝tanα補充：6×9＝54種的表格以及推導(dǎo)方法(定名法則和定號法f(β)sicotacosecs→nβsβnβtβcβcβf↘β↓360°ksinαcosαtanαcotαsecαcscα90°-cosαsinαcotαtanαcscαsecαα3實用精品文檔α-sinαcotα-tanαcscααcosα-tanαcotα-secαα-sinαcosα-secααcosα-sinαcscααcosαcotα-tanαα-sinα-tanαcotα-sinα-tanαcotα定名法則90°的奇數(shù)倍+α的三角函數(shù)，其絕對值與α三角函數(shù)的絕對值互為。90°的偶數(shù)倍+α的三角函數(shù)與α的三角函數(shù)絕對值相同。也就是“奇余偶同，奇變偶不變”定號法則13實用精品文檔將α看做銳角(注意是“看做”)，按所得的角的象限，取三角函數(shù)的符號。也就是“象限定號，符號看象限”.(或為“奇變偶不變，符號看象限”2在Kπ/中如果K為奇數(shù)時函數(shù)名不變，若為偶數(shù)時函數(shù)名于正負號有可口訣；一全二正弦，三切四余弦，即第一象限全部為正，第二象限角正弦為正，第三為正切為正，第四象限余弦為正。)比如:90°+α。定名：90°是90°的奇數(shù)倍，所以應(yīng)取余函數(shù)；定號：將α看做銳角，那么90°+α是第二象限角，第二象限角的正弦為正，余弦為負。所以sin(90°+α)=cosα,cos(90°+α)＝-sinα這個非常神奇，屢試不爽~還有一個口訣“縱變橫不變，符號看象限”，例如：sin(90°+α)，90°的終邊在縱軸上，所以函數(shù)名變?yōu)橄喾吹暮瘮?shù)名，即cos，將α看做銳角，那么90°+α是第二象限角，第二象限角的正弦為正，所以sin(90°+α)=cosαtan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)檔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]nsincostancotcos(2α)=(cosα)^2-(sinα)^2=2(cosα)^2-1=1-2(sinα)^2tan(2α)=2tanα/(1-tan^2α)sin(3α)=3sinα-4sin^3α=cos(3α)=4cos^3α-3cosα=tan(3α)=(3tanα-tan^3α)/(1-3tan^2α)=tanαtan(π/3+α)tan(π/3-α)13實用精品文檔tan(α/2)=±√((1-cosα)/(1+cosα))=sinα/(1+cosα)=(1-cosα)/sinαAsinα+Bcosα=√(A^2+B^2)sin(α+φ)(tanφ=B/A)Asinα-Bcosα=√(A^2+B^2)cos(α-φ)(tanφ=-A/B)sin(a)=(2tan(a/2))/(1+tan^2(a/2))cos(a)=(1-tan^2(a/2))/(1+tan^2(a/2))tan(a)=(2tan(a/2))/(1-tan^2(a/2))sin^2α=(1-cos(2α))/2=versin(2α)/2cos^2α=(1+cos(2α))/2=covers(2α)/2tan^2α=(1-cos(2α))/(1+cos(2α))sinα=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)]a*sin(a)+b*cos(a)=sqrt(a^2+b^2)sin(a+c)[其中，tan(c)=b/a]a*sin(a)-b*cos(a)=sqrt(a^2+b^2)cos(a-c)[其中，tan(c)=a/b]1+sin(a)=(sin(a/2)+cos(a/2))^21-sin(a)=(sin(a/2)-cos(a/2))^2其他非重點三角函數(shù)csc(a)=1/sin(a)sec(a)=1/cos(a)cos30=sin60sin30=cos60tanα+cotα=2/sin2αtanα-cotα=-2cot2α1+cos2α=2cos^2α1-cos2α=2sin^2α1+sinα=[sin(α/2)+cos(α/2)]^2sinα+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)=0cosx+cos2x+...+cosnx=[sin(n+1)x+sinnx-sinx]/2sinx左邊=2sinx(cosx+cos2x+...+cosnx)/2sinx=[sin2x-0+sin3x-sinx+sin4x-sin2x+...+sinnx-sin(n-2)x+sin(n+1)x-sin(n-1)x]/2sinx(積化和差)=[sin(n+1)x+sinnx-sinx]/2sinx=右邊等式得證sinx+sin2x+...+sinnx=-[cos(n+1)x+cosnx-cosx-1]/2sinx左邊=-2sinx[sinx+sin2x+...+sinnx]/(-2sinx)=[cos2x-cos0+cos3x-cosx+...+cosnx-cos(n-2)x+cos(n+1)x-cos(n-1)x]/(-2sinx)=-[cos(n+1)x+cosnx-cosx-1]/2sinx=右邊等式得證三倍角公式推導(dǎo)sin3a=sin(2a+a)=sin2acosa+cos2asina=2sina(1-sin^2a)+(1-2sin^2a)sina=3sina-4sin^3acos3a=cos(2a+a)=cos2acosa-sin2asina=(2cos^2a-1)cosa-2(1-cos^2a)cosa=4cos^3a-3cosasin3a=3sina-4sin^3a=4sina(3/4-sin^2a)=4sina[(√3/2)^2-sin^2a]=4sina(sin^260°-sin^2a)=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=4cos^3a-3cosa=4cosa(cos^2a-3/4)=4cosa(cos^2a-cos^230°)=4cosa(cosa+cos30°)(cosa-cos30°)=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)