三角函数关系式大全

同角三角函数关系式三三角函数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实用精品文档sinα/cosα＝tanα＝secα/cscαcosα/sinα＝cotα＝cscα/secαABCA角A的对边比斜边,A正切等于对边比邻边,设α为任意角，终边相同的角的同一三角函数的值相等：精品文档三三角函数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实用精品文档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实用精品文档cot(3π/2＋α)＝－tanαsin(3π/2－α)＝－cosαcos(3π/2－α)＝－sinαtan(3π/2－α)＝cotαcot(3π/2－α)＝tanα补充：6×9＝54种的表格以及推导方法(定名法则和定号法f(β)sicotacosecs→nβsβnβtβcβcβf↘β↓360°ksinαcosαtanαcotαsecαcscα90°-cosαsinαcotαtanαcscαsecαα3实用精品文档α-sinαcotα-tanαcscααcosα-tanαcotα-secαα-sinαcosα-secααcosα-sinαcscααcosαcotα-tanαα-sinα-tanαcotα-sinα-tanαcotα定名法则90°的奇数倍+α的三角函数，其绝对值与α三角函数的绝对值互为。90°的偶数倍+α的三角函数与α的三角函数绝对值相同。也就是“奇余偶同，奇变偶不变”定号法则13实用精品文档将α看做锐角(注意是“看做”)，按所得的角的象限，取三角函数的符号。也就是“象限定号，符号看象限”.(或为“奇变偶不变，符号看象限”2在Kπ/中如果K为奇数时函数名不变，若为偶数时函数名于正负号有可口诀；一全二正弦，三切四余弦，即第一象限全部为正，第二象限角正弦为正，第三为正切为正，第四象限余弦为正。)比如:90°+α。定名：90°是90°的奇数倍，所以应取余函数；定号：将α看做锐角，那么90°+α是第二象限角，第二象限角的正弦为正，余弦为负。所以sin(90°+α)=cosα,cos(90°+α)＝-sinα这个非常神奇，屡试不爽~还有一个口诀“纵变横不变，符号看象限”，例如：sin(90°+α)，90°的终边在纵轴上，所以函数名变为相反的函数名，即cos，将α看做锐角，那么90°+α是第二象限角，第二象限角的正弦为正，所以sin(90°+α)=cosαtan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)档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实用精品文档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其他非重点三角函数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左边=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(积化和差)=[sin(n+1)x+sinnx-sinx]/2sinx=右边等式得证sinx+sin2x+...+sinnx=-[cos(n+1)x+cosnx-cosx-1]/2sinx左边=-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=右边等式得证三倍角公式推导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°)