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Relaciones entre las funciones trigonomtricas.

ctg x= 1tg x sec x=1cos x csc x=1sen x tg x=sen xcos x ctg

x=cos xsen x

sen2 x+cos2x=1 ; 1+tg2x=sec2x ; 1+ctg2x=csc2x.

Funciones trigonomtricas de (x+y) y (x-y).

sen (x+y)= senx cosy + cosx sen y. sen (x-

y)=senx cosy - cosx seny.

cos(x+y)=cosx cosy - sen x sen y. cos(x-y)= cosx

cosy +senx seny.

tg(x+y)= tgx + tgy1-tgx tgy. tg(x-

y)=tgx -tgy1+tg x tg y.

Funciones trigonomtricas de 2x y de x.

sen 2x=2 sen x cos x; cos 2x=cos2 x - sen2 x;

tg 2x=2 tg x1 -tg2x.

sen x2= 1-cos x2; cos x2=1+cos x2; tg

x2=1- cos x1+cosx.

sen2x=1/2-1/2 cos 2x; cos2x=1/2+1/2 cos 2x.

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Derivadas.

I dcdx=0. XIIddx(uv)=vuv-

1dudx+lnu.uvdvdx

II dxdx=1. XIIIddx(senv)=cosv

dvdx.

IIIddx(u+v-w)=dudx+dvdx-dwdx XIVddx(cosv)=-sen vdvdx.

IVddx(cv)=cdvdx. XVddx(tgv)=sec2vdvdx.

V ddx(uv)= udvdx+v dudx. XVIddx(ctgv)=-

csc2vdvdx.

VI ddx(vn)=nvn-1 dvdx . XVIIddx(secv)=secv.

tgvdvdx.

VIa ddx(xn)=nxn-1. XVIII ddx(cscv)=-

cscv.ctgvdvdx.

VII ddx(uv)=vdudx-udvdxv2. XIXddxvers v=sen

vdvdx

VIIa ddx(uc)=dudxc. XXddx(arc

senv)=dvdx1-v2.

VIII dydx=dydv.dvdx, siendo y funcion de v. XXIddx(arc cosv)=-

dvdx1-v2

IXdydx=1dxdy, siendo y funcion de x. XXIIddx(arc tgv)=

dvdx1+v2

Xddx(lnv)=dvdxv=1vdvdx. (lnv=logev ) XXIIIddx(arc ctgv)=-

dvdx1+v2

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Xaddx(logv)=log evdvdx. XXIVdvdx(arc sec v)=

dvdxvv2-1

XI ddx(av)=avln advdx. XXV ddx(arc cscv)=

-dvdxvv2-1

XIaddx(ev)=evdvdx XXVIddx(arc vers

v)=dvdx2v-v2

Integrales

1 (du+dv-dw)=du+dv-dw 19 dvv2-a2=12alnv-av+a+c(v2>a2)

2 adv=adv 19a dva2-v2=12alna+va-v+c

(v2

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15 ctgv dv=ln senv+c

16 sec v dv=ln(secv+tgv)+c

17csc v dv=ln(cscv-ctgv)+c

18 dvv2+a2=1aarctgva+c