HPHP48-E-@§,*“
*UBTrigonométrie circulaire

**UTransformationsU:

cos(-x)=cos(x)
sin(-x)=-sin(x)
tan(-x)=-tan(x)
cotan(-x)=-cotan(x)

cos(x+‡)=-cos(x)
sin(x+‡)=-sin(x)
tan(x+‡)=tan(x)
cotan(x+‡)=cotan(x)

cos(‡-x)=-cos(x)
sin(‡-x)=sin(x)
tan(‡-x)=-tan(x)
cotan(‡-x)=-cotan(x)

cos(x+‡/2)=-sin(x)
sin(x+‡/2)=cos(x)
tan(x+‡/2)=-1/tan(x)
cotan(x+‡/2)=-tan(x)

cos(‡/2-x)=sin(x)
sin(‡/2-x)=cos(x)
tan(‡/2-x)=1/tan(x)
cotan(‡/2-x)=tan(x)

**UAdditionU:

cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
sin(a+b)=sin(a)cos(b)+cos(a)sin(b)

tan(a+b)=Utan(a)+tan(b)
         1-tan(a)tan(b)

cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
sin(a-b)=sin(a)cos(b)-sin(b)cos(a)

tan(a-b)=Utan(a)-tan(b)
         1+tan(a)tan(b)

**UDuplicationU:

cos(2x)=cos²(x)-sin²(x)
       =*cos²(x)-1
       =1-2sin²(x)
sin(2x)=2sin(x)*cos(x)

tan(2x)= U 2tanx 
         1-tan²x

cos²x=(1+cos2x)/2
sin²x=(1-cos2x)/2

tan²x=U1-cos(2x)
      1+cos(2x)

1+tan²x=1/cos²(x)

cos(3x)=4cos³(x)-3cos(x)
sin(3x)=-4sin³(x)+3sin(x)

**UExpression en fonction
  Ude tan x/2:

  t=tan x/2
sin(t)=2t/1+t²
cos(t)=1-t²/1+t²
tan(t)=2t/1-t²

**UFormules de transformationU:

cos(p)+cos(q)=2cos(p+q)/2*cos(p-q)/2
cos(p)-cos(q)=-2sin(p+q)/2*sin(p-q)/2
sin(p)+sin(q)=2sin(p+q)/2*cos(p-q)/2
sin(p)-sin(q)=2sin(p-q)/2*cos(p+q)/2

cos(a)cos(b)=[cos(a+b)+cos(a-b)]/2
sin(a)sin(b)=[cos(a+b)-cos(a-b)]/2
sin(a)cos(b)=[sin(a+b)+sin(a-b)]/2


*UBTrigonométrie inverse

**UArcsinusU:

Arcsin [-1,1][-‡/2,‡/2]

xÒ[‡/2,‡/2] Arcsin(sin x)=x

xÒ[-1,1] sin(Asin x)=x
         Asin(-x)=-Asinx
         cos(Asinx)=ƒ(1-x²)
xÒ]-1,1[ tan(Asinx)=x/ƒ(1-x²)

ˆ(Asinx)=1/ƒ(1-x²)

**UArccosinusU:

Arccos [-1,1][0,‡]

xÒ[0,‡] Acos(cos)=x

xÒ[-1,1] Acos(cosx)=x
         Acos(-x)=‡-Acosx
         sin(Acosx)=ƒ(1-x²)
         tan(Acosx)=ƒ(1-x²)/x

Acosx+Asinx=‡/2
ˆ(Acosx)=-1/ƒ(1-x²)

**UArctangenteU:

Arctan ]-‡/2,.[

xÒ]-‡/2,‡/2[ Atan(tan)=x

xÒ tan(Atanx)=x
    Atan(-x)=-Atanx
    cos(Atanx)=1/ƒ(1+x²)
    sin(Atanx)=x/ƒ(1+x²)

xÒ4+1 Atanx+Atan1/x=‡/2
xÒ4-1              -‡/2
ˆ(Atanx)=1/(1+x²)
ˆ(Atanu)=u\ˆ(1+u²)

**UCotangenteU:

]-‡/2,‡/2[

ˆCotanx=-1/sin²x=-1-cotan²x

**UArccotangenteU:

R[0,‡]
xÒ]0,‡[ Arcotan(cotan x)=x

xÒ cotan(arccotanx)=x
    Arctan(-x)=‡-Arccotan(x)

ˆArccotan=-1/1+x²
xÒ Arccotanx+Artanx=‡/2

*UBTrigonométrie hyperbolique

**UAdditionU:

Ch(a+b)=ChaChb+ShaShb
Sh(a+b)=ShaChb+ShbCha

Th(a+b)=U Tha+Thb
        1+ThaThb

Ch(a-b)=ChaChb-ShaShb
Sh(a-b)=ShaChb-ShbCha

Th(a-b)=U Tha-Thb
        1-ThaThb

**UDuplicationU:

Ch2x=Ch^2+Sh^2
    =2Ch^2-1
    =1-2Sh^2
Sh2x=2ShxChx
Th2a=2Tha/1+Th^2

**UTransformationsU:

Chp+Chq=2*Ch((p+q)/2)*Ch((p-q)/2)
Chp-Chq=2*Sh((p+q)/2)*Sh((p-q)/2)
Shp+Shq=2*Sh((p+q)/2)*Ch((p-q)/2)
Shp-Shq=2*Ch((p+q)/2)*Sh((p-q)/2)

ChaChb=1/2[Ch(a+b)+Ch(a-b)]
ShaShb=1/2[Ch(a+b)-Ch(a-b)]
ShaChb=1/2[Sh(a+b)+Sh(a-b)]

**UExpression en fonction
  Ude t=Thx/2:

Shx=2t/1-t^2
Chx=1+t^2/1-t^2
Thx=2t/1+t^2

**UAutres formulesU:

Chnx=U(Chx+Shx)^n+(Chx-Shx)3Un
                2

Shnx=U(Chx+Shx)^n-(Chx-Shx)3Un
                2

*UBTrigonométrie hyperbolique
 UBinverse

xÒ  Argsh(x)=LN(x+ƒ(x^2+1))
xÒ[1,Ÿ[  Argch(x)=LN(x +ƒ(x^2-1))
xÒ]-1,1[  Argth(x)=1/2LN((1+x)/(1-x))
xÒ]-Ÿ,-1[Ñ]1,Ÿ]  Argcoth(x)=1/2LN((1+x)/(x-1))
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