三角函数、导数、微分、积分

1、三角函数诱导公式 tgA=tanA =sin acosasin( a)sin a sin(-a) = sin(+a) = sin( -a)= sin( +a)=22cosacosasina-sinacos(-a) = cosacos(-a) = cos(+a) = cos( -a)= cos( +a)=22sina-sina-cosa-cosa两角和差公式sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-cosAsinBcos(A+B) = cosAcosB-sinAsinBcos(A-B) = cosAcosB+sinAsinBtan(A+B)

2、 =tanAtanBcotAcotB -1cot(A+B) =tan(A-B) =1- tanAtanBtanAtanB1tanAtanBcot(A-B) =cotBcotAcotAcotB1cotBcotA倍角公式三倍角公式半角公式tan2A =2tanA1 tan2 ASin2A=2SinA?CosAsin3A = 3sinA-4(sinA)3A )=1cos Asin(22cos3A = 4(cosA) 3-3cosAA )=1cos Acos(22Cos2A=tan3a=A )=1cos Atan(2222Atana tan( +a) tan(-a)21cosACos A-SinA=2

3、CosA-1=1-2sin33cot(A )=1cos A21cosAtan(A )= 1cos A =sin A2sin A1cos A和差化积积化和差sina+sinb=2sinab cos absinasinb = -1 cos(a+b)-cos(a-b)222sina-sinb=2cosab sin abcosacosb =1cos(a+b)+cos(a-b)222cosa+cosb = 2cosab cos absinacosb =1sin(a+b)+sin(a-b)222cosa-cosb = -2sinab sin abcosasinb =1sin(a+b)-sin(a-b)22

4、2tana+tanb= sin( ab)cosa cosb2 tan asina=21 (tan a )2 21csc(a) =sin acot 2 a1csc2 aa-ae - esinh(a)=万能公式1(tan a ) 22 tan acosa=2tana=2(tan a ) 2(tan a )21122其他非重点三角函数sec(a) =1cosa1tan 2 a 1sec2 acos2 a双曲函数a-asinh( a)cosh(a)=eetg h(a)=cosh(a)2等价无穷小sin x xarcsin x x arctanxln(1 x) xex1 xtan x x2(1 x) u

5、1 uxn 1 x 1 xax1 x ln a1 cos x xn2两个重要的极限lim sin x1lim (11 ) xex 0xxx导数、微分、积分函数的和差积商求导函数的和差积商微分法函数的和差积商求导法则法则则u vuvd (u v) du dvxu 1c(u1)xu dxu1Cuuvuvd (Cu )Cdukfx dxk fx dxCuuvuvd (uv)vduudvfxg x dxfx dxg x dxvuvd (u )vduudvfx x dxfu duuvv2uxv 2高阶导数函数 yf ( x) 的导数 y f ( x) 称为一阶导数，记作y 或 dy；把 y f ( x)

6、 的导数称为dx二阶导数， 记作 yy d2 yddy；类似的， 二阶导数的导数称为三阶导数；三或dxdxdx2d n y阶导数的导数称为四阶导数；（n 1）导数的导数叫做n 阶导数记作dx n导数公式C 0( x u ) uxu 1( 1 ) 1xx 2(sin x) cos x(cos x) sin x(tan x) sec2 x微分公式dy f ( x) dxd( xu )ux u 1dxd ()dxd(sin x)cos xdxd (cos x)sin xdxd(tan x)sec2 xdx积分公式kdxkx Cx u dxxu1Cu11 dxln xCxcos xdxsin xCsi

7、n xdxcosxC1 dx sec2 xdx tan x C cos2 xtan xdxln cosxC2212(cot x)csc xd(cot x)cscxdxsin 2 x dxcsc xdxcot xCcot xdxln sin xC(sec x) sec x tan x(csc x) csc x cot x( a x ) a x ln a(ex ) ex(log ax) 1x ln a(ln x)1x(arcsin x) 1x 21(arccos x) 11x 2(arctan x) 11 x 2( arc cot x) 11x 2( shx) chx(chx)shx(thx) 1

8、ch 2 x1(arshx )(archx )1x 21d (sec x)sec x tan xdxd (csc x)csc x cot xdxd(a x )ax ln adxd(ex )ex dxd(log ax)1dxx ln ad (ln x)1 dxxd(arcsin x)1dx1x 2d(arccos x)1dx1x 2d(arctan x)1dx1x 2d(arc cot)1dx1x 2d ()dxsecxdxln secxtan xCsecx tan xdxsecxCcscxdxln cscxcot xCcsc x cot xdxcsc xCa xdxaxCln aex dxexCdxC1 dxln xCx1dxarcsin xC1x2dxC11x2 dxarctan xCdxC