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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
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