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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 48 "CorrigŽ du TD 2 : Courbes
, plans, surfaces, etc." }}{PARA 256 "" 0 "" {TEXT -1 25 "Jean SŽbasti
en ROY, 1998." }}{PARA 0 "" 0 "" {TEXT -1 71 "Note : Les erreurs les p
lus frŽquentes proviennent de l'oubli de faire " }{TEXT 256 7 "restart
" }{TEXT -1 28 " et de recharger le package " }{TEXT 258 5 "plots" }
{TEXT -1 6 " avec " }{TEXT 257 10 "with(plots" }{TEXT -1 19 ") apr�s u
n restart." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Ex 1 : Plans tange
nts ˆ un ellipso•de" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar
t:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Comme on va avoir besoin de
 quelques fonctions des packages '" }{TEXT 259 5 "plots" }{TEXT -1 6 "
' et '" }{TEXT 260 6 "linalg" }{TEXT -1 16 "' on les charge." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "with(plots):with(linalg):" }
}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Le vecteur directeur de la droite \+
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "vDroite:=[1,3,-2];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(vDroiteG7%\"\"\"\"\"$!\"#" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Et une Žquation paramŽtrique de ce
lle-ci (" }{TEXT 261 5 "evalm" }{TEXT -1 83 " (m pour 'matrices'), per
met entre autres de multiplier un vecteur par un scalaire)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "eDroite:=evalm(t*vDroite);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(eDroiteG-%'vectorG6#7%%\"tG,$F)\"\"
$,$F)!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Les options graphiqu
es communes ˆ tous les graphes :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 62 "opts:=view=[-2..2,-2..2,-2..2],scaling=constrained,ax
es=boxed;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%optsG6%/%%viewG7%;!\"#
\"\"#F)F)/%(scalingG%,constrainedG/%%axesG%&boxedG" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 27 "Le graphique de la droite :" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 54 "droite:=spacecurve(eDroite,t=-2..2,opts,shadi
ng=none):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Et celui de l'ellips
o•de, tracŽ en coordonnŽes cylindriques apr�s avoir ŽtŽ paramŽtrŽ :" }
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db]n$\"+--^vZFjgcl$!+oMn$=$Fjgcl7%$\"+(QpMn\"Fjgcl$\"+h\"3/-&Fjgcl$!+u
(QpM$Fjgcl7%$\"+S?5b<Fjgcl$\"+?hIl_Fjgcl$!+!3/-^$Fjgcl7%$\"+$pMn$=Fjgc
l$\"+zS?5bFjgcl$!+'QpMn$Fjgcl7%$\"+YtO=>Fjgcl$\"+Q?5bdFjgcl$!+#pMn$QFj
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%%NONEG-%(SCALINGG6#%,CONSTRAINEDG-%*AXESSTYLEG6#%$BOXG-%%VIEWG6%;Fid
\\n$\"\"#F)F_[^nF_[^n" 1 2 0 1 0 2 1 1 2 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "L'Žquation de l'el
lipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eEllipsoide:=
x^2+y^2+2*z^2-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,eEllipsoideG,**
$)%\"xG\"\"#\"\"\"\"\"\"*$)%\"yGF)F*F+*$)%\"zGF)F*F)!\"\"F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Vecteur normal en tout point de l'
ellipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "grad(eEllip
soide,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7%,$%\"
xG\"\"#,$%\"yGF),$%\"zG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "
On cherche ˆ trouver tous les (x,y,z) tels que ce vecteur soit colinŽa
ire au vecteur directeur de la droite, cad que le produit vectoriel de
s deux vecteur soit nul." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 
"crossprod(vDroite,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#
7%,&%\"zG\"#7%\"yG\"\"%,&%\"xG!\"%F(F.,&F*\"\"#F-!\"'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 54 "Il faut rŽsoudre le syst�me prŽcŽdent (=[
0,0,0]) mais " }{TEXT 266 5 "solve" }{TEXT -1 20 " n'accepte que des '
" }{TEXT 262 3 "set" }{TEXT -1 14 "' et non des '" }{TEXT 263 6 "vecto
r" }{TEXT -1 22 "', donc on convertit :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "convert(%,set);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%
,&%\"xG!\"%%\"zGF&,&F'\"#7%\"yG\"\"%,&F*\"\"#F%!\"'" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 24 "Et on rŽsoud en x et y :" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "solve(%,\{x,y\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#<$/%\"xG,$%\"zG!\"\"/%\"yG,$F'!\"$" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "On enl�ve les accolades pour utiliser le resultat \+
dans un '" }{TEXT 264 4 "subs" }{TEXT -1 12 "' plus bas :" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "solution:=op(%);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%)solutionG6$/%\"xG,$%\"zG!\"\"/%\"yG,$F)!\"$" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "On substitue la solution dans l'Žq
uation de l'ellipso•de :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"subs(solution,eEllipsoide);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%
\"zG\"\"#\"\"\"\"#7!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 
"Et on rŽsoud en " }{TEXT 267 1 "z" }{TEXT -1 119 " pour trouver tous \+
les points de l'ellipso•de en lesquels la normale est colinŽaire au ve
cteur directeur de la droite :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "solutionZ:=solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*sol
utionZG6$,$*$-%%sqrtG6#\"\"$\"\"\"#\"\"\"\"\"',$F'#!\"\"F/" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "Les valeurs des points en question :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "subs(solution,z=solutionZ[1]
,[x,y,z]);subs(solution,z=solutionZ[2],[x,y,z]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7%,$*$-%%sqrtG6#\"\"$\"\"\"#!\"\"\"\"',$F%#F,\"\"#,$F%#
\"\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,$*$-%%sqrtG6#\"\"$\"\"
\"#\"\"\"\"\"',$F%#F,\"\"#,$F%#!\"\"F-" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 165 "On cherche alors l'Žquation des plans perpendiculaires ˆ
 la droite, cad l'ensemble des points M(x,y,z) tq OM . vDroite = const
ante. On calcule plan0 = OM . vDroite :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "plan0:=dotprod([x,y,z],vDroite);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&plan0G,(%\"xG\"\"\"%\"yG\"\"$%\"zG!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 122 "Les plans doivent passer par les points \+
prŽcŽdement trouvŽs, d'o� les valeurs des  constantes et les Žquations
 des plans :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plan:= i ->
 plan0-subs(solution,z=solutionZ[i],plan0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%planGR6#%\"iG6\"6$%)operatorG%&arrowGF(,&%&plan0G\"
\"\"-%%subsG6%%)solutionG/%\"zG&%*solutionZG6#9$F-!\"\"F(F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Par exemple :" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "plan(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
*%\"xG\"\"\"%\"yG\"\"$%\"zG!\"#*$-%%sqrtG6#F'\"\"\"\"\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 40 "On trace les plans sans les parametrer :
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lan(1),x=-2..2,y=-2..2,z=-2..2,grid=[13,13,13],style=patchnogrid,opts)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "plan2:=implicitplot3d(
plan(2),x=-2..2,y=-2..2,z=-2..2,grid=[13,13,13],style=patchnogrid,opts
):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "RŽsultat final :" }}}
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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Ex 3 : Intersection de deux surfa
ces" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Les Žquations des deux surfac
es :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "e1:=y^2*(x^2+z^2)-(
c^2+b^2)*x^2-(c^2-a^2)*z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G,
(*&)%\"yG\"\"#\"\"\",&*$)%\"xGF)F*\"\"\"*$)%\"zGF)F*F/F/F/*&,&*$)%\"cG
F)F*F/*$)%\"bGF)F*F/F/F-F*!\"\"*&,&F5F/*$)%\"aGF)F*F;F/F1F*F;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "e2:=x^2/a^2+y^2/(c^2)-z^2/b^
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20 "6#>%$cstG6%/%\"aG$\"\"&!\"\"/%\"bG\"\"#/%\"cG\"\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "Pour obtenir l'Žquation avec les constant
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "C1:=implicitplot3d(subs(cst,
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}{TEXT -1 4 " et " }{TEXT 268 2 "e2" }{TEXT -1 25 ". On commence par i
soler " }{TEXT 270 3 "y^2" }{TEXT -1 4 " de " }{TEXT 271 3 "e1 " }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "ey2:=solve
(e1,y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ey2G,$*&,**&)%\"xG\"\"
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F+F,\"\"\"F,,&*$F)F,F9*$F4F,F9!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "Que l'on injecte dans " }{TEXT 272 2 "e2" }{TEXT -1 2 " :
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "e2b:=factor(subs(y^2=ey
,e2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$e2bG*&,**()%\"xG\"\"#\"\"
\")%\"cGF*F+)%\"bGF*F+\"\"\"*(%#eyGF0)%\"aGF*F+F.F+F0*()%\"zGF*F+F3F+F
,F+!\"\"*(F3F+F,F+F.F+F8F+*()F4\"\"#F+)F-\"\"#F+)F/\"\"#F+!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "On cherche " }{TEXT 273 1 "x" }
{TEXT -1 2 ", " }{TEXT 274 1 "z" }{TEXT -1 4 " tq " }{TEXT 275 5 "e2b=
0" }{TEXT -1 31 ". cad tel que le numŽrateur de " }{TEXT 276 3 "e2b" }
{TEXT -1 22 " soit nul (notons que " }{TEXT 277 3 "e2b" }{TEXT -1 24 "
 n'est pas definie pour " }{TEXT 278 5 "x=z=0" }{TEXT -1 1 ")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "numer(e2b);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,**()%\"xG\"\"#\"\"\")%\"cGF'F()%\"bGF'F(\"\"\"*(%
#eyGF-)%\"aGF'F(F+F(F-*()%\"zGF'F(F0F(F)F(!\"\"*(F0F(F)F(F+F(F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Le troisi�me terme est " }{TEXT 
279 3 "> 0" }{TEXT -1 7 " donc :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "expand((x*b-z*a)*(x*b+z*a));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&)%\"xG\"\"#\"\"\")%\"bGF'F(\"\"\"*&)%\"zGF'F()%\"aG
F'F(!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "... est ce que l'on \+
cherche ˆ rŽsoudre. Isolons " }{TEXT 280 3 "x^2" }{TEXT -1 2 " :" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "solve(%,x^2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&*&)%\"zG\"\"#\"\"\")%\"aGF'F(F(*$)%\"bG\"\"#F(
!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "D'ou " }{TEXT 281 3 "y^2
" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "factor
(subs(x^2=%,ey2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)%\"cG\"\"#\"
\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Resumons : les solutions \+
sont des droites du type : x=+/- z a/b , y=c" }}{PARA 0 "" 0 "" {TEXT 
-1 17 "On en trace une :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
127 "D1:=spacecurve(subs(cst,[-z*a/b,c,z]),z=-5..5,view=[-5..5,-5..5,-
5..5],scaling=constrained,shading=none,color=red,thickness=3):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Et des coupes bien chosies des gra
phes des surfaces :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "C1P:
=implicitplot3d(subs(cst,e1),x=-5..5,subs(cst,y=-c..c),z=-5..5,opts,sh
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implicitplot3d(subs(cst,e2),x=-5..5,subs(cst,y=-c..c),z=-5..5,opts,sha
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0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "fx:=t->cos(t)^3;fy:=t-
>sin(t)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxGR6#%\"tG6\"6$%)ope
ratorG%&arrowGF(*$)-%$cosG6#9$\"\"$\"\"\"F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#fyGR6#%\"tG6\"6$%)operatorG%&arrowGF(*$)-%$sinG6#9$
\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Une tangente
 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "-(y-fy(t))*D(fx)(t)+(
x-fx(t))*D(fy)(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(,&%\"yG\"\"
\"*$)-%$sinG6#%\"tG\"\"$\"\"\"!\"\"F')-%$cosGF,\"\"#F/F*F'F.*(,&%\"xGF
'*$)F2F.F/F0F')F*F4F/F2F'F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*()-%$cosG6#%\"t
G\"\"#\"\"\"-%$sinGF(\"\"\"%\"yGF.\"\"$*$F%F+!\"$*$)F&\"\"%F+F0*&F&F.%
\"xGF.F0*&)F&F0F+F7F+F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
tg:=unapply(%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tgGR6#%\"tG6\"
6$%)operatorG%&arrowGF(,,*()-%$cosG6#9$\"\"#\"\"\"-%$sinGF1\"\"\"%\"yG
F7\"\"$*$F.F4!\"$*$)F/\"\"%F4F9*&F/F7%\"xGF7F9*&)F/F9F4F@F4F;F(F(F(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Condition pour que deux tangente
s soient orthogonales : (" }{TEXT 282 1 "D" }{TEXT -1 31 " est l'opŽra
teur de dŽrivation)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "D(fx
)(t)*D(fx)(u)+D(fy)(t)*D(fy)(u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
**)-%$cosG6#%\"tG\"\"#\"\"\"-%$sinGF(\"\"\")-F'6#%\"uGF*F+-F-F1F.\"\"*
**)F,F*F+F&F.)F3F*F+F0F.F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "On \+
simplifie un peu :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "facto
r(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*,-%$sinG6#%\"tG\"\"\"-%$co
sGF'F)-F&6#%\"uGF)-F+F-F),&*&F/\"\"\"F*F2F2*&F%F2F,F2F2F2\"\"*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "%/(9*sin(t)*cos(t)*sin(u)*co
s(u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$cosG6#%\"uG\"\"\"-F&6
#%\"tGF)F)*&-%$sinGF+F)-F/F'F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "combine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cos
G6#,&%\"uG\"\"\"%\"tG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
7 "eq1:=%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G-%$cosG6#,&%\"uG
\"\"\"%\"tG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Et on rŽsoud \+
le systeme : deux tangentes, qui se coupent, et sont orthogonales :" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "\{tg(u),tg(t),eq1\};" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#<%,,*()-%$cosG6#%\"tG\"\"#\"\"\"-%$sin
GF)\"\"\"%\"yGF/\"\"$*$F&F,!\"$*$)F'\"\"%F,F1*&F'F/%\"xGF/F1*&)F'F1F,F
8F,F3-F(6#,&%\"uGF/F*!\"\",,*()-F(6#F>F+F,-F.FDF/F0F,F1*$FBF,F3*$)FCF6
F,F1*&FCF/F8F,F1*&)FCF1F,F8F,F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "s:=solve(%,\{x,y,u\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"
sG6$<%/%\"uG,&%\"tG\"\"\"%#PiG#!\"\"\"\"#/%\"yG,(-%$sinG6#F*F+*$)F3\"
\"$\"\"\"F.*&-%$cosGF5F+)F3F/F9F./%\"xG,(F:F+F3F+F6F.<%/F(,&F*F+F,#F+F
//F1F@/F?,(F:F+F6F+F3F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Les de
ux solutions sont les m�mes. Prenons en une :" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "s[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"uG
,&%\"tG\"\"\"%#PiG#!\"\"\"\"#/%\"yG,(-%$sinG6#F'F(*$)F0\"\"$\"\"\"F+*&
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