Relaciones entre las funciones trigonométricas.
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 trigonométricas 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 trigonométricas de 2x 2x y de ½ 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 . sen2 x=1/2-1/2
cos 2x;
cos2 x=1/2+1/2
cos 2x.
Derivadas. I dcdx=0.
XIIddx(uv)=vuv-
1dudx+lnu.uvdvdx
II dxdx=1.
XIIIddx(senv)=cosv
dvdx.
IIIddx(u+v-w)=dudx+dvdx-dwdx
ddx(cosv)=-sen v dvdx. dvdx. XIV ddx(cosv)=
IV ddx(cv)=cdvdx.
ddx(tgv)=sec2vdvdx. XV ddx(tgv)=sec2vdvdx.
V ddx(uv)= udvdx+v dudx.
XVIddx(ctgv)=-
csc2vdvdx.
VI ddx(vn)=nvn-1 dvdx .
XVIIddx(secv)=secv.
dvdx. tgv dvdx
VIa ddx(xn)=nxn-1.
XVIII ddx(cscv)=-
cscv.ctgvdvdx.
VII ddx(uv)= vdudx-udvdxv2.
ddxvers v=sen XIX ddxvers
vdvdx
VIIa ddx(uc)=dudxc.
XX ddx(arc
senv)=dvdx1-v2.
VIII dydx=dydv.dvdx, siendo y funcion de v.
XXI ddx(arc cosv)=-
dvdx1-v2
IX dydx=1dxdy, siendo y funcion de x.
XXII ddx(arc tgv)=
dvdx1+v2
X ddx(lnv)=dvdxv=1vdvdx. (lnv=logev dvdx1+v2
)
ddx(arc XXIII ctgv)=-
Xaddx(logv)=log evdvdx.
dvdx(arc sec v)= XXIV dvdx(arc
dvdxvv2-1
XI ddx(av)=avln advdx.
XXV ddx(arc cscv)=
-dvdxvv2-1
XIaddx(ev)=evdvdx
XXVIddx(arc vers
v)=dvdx2v-v2
Integrales v2>a2 ) 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
3 dx=x+c
20 dva2-v2=arcsenva+c
4 vndv=vn+1n+1+c 21dvv2±a2=ln(v+v2±a2)+c 5 dvv=ln v+c 6 avdv=avln a+c
22a2-v2dv=v2a2-v2a22arcsenva+c 23 v2±a2dv=v2v2±a2±a22ln(v+v2±a2)
7 evdv=ev+c 8senv dv=-cosv+c 9 cosv dv=senv+c 10 sec2v dv=tgv+c 11 csc2v dv=-ctgv+c 12 secv tgv dv=secv+c 13 cscv ctgv dv=-cscv+c 14 tgv dv=-ln cosv+c=ln secv+c
15 ctgv dv=ln senv+c dv=ln(secv+tgv)+c 16 sec v dv=ln(secv+tgv)+c d v=ln(cscv-ctgv)+c +c 17csc v dv=ln(cscv-ctgv)
18 dvv2+a2=1aarctgva+c
