A+B+C= pi

Sujet: A+B+C= pi Dim 25 Jan 2009, 19:30

salam

ABC un triangle.

montrer que:

1) sinA.cosA + sinB.cosB + sinC.cosC = 2.sinA.sinB.sinC

2) sinA/sinBsinC + sinB/sinAsinC + sinC/sinAsinB =

2( cotanA + cotanB + cotanC)

Sujet: Re: A+B+C= pi Dim 25 Jan 2009, 21:39

salut houssa
1)sinA.cosA+sinB.cosB+sinC.cosC=1/2(sin2A+sin2B)+sinC.cosC
=1/2(2sin(A+B).cos(A-B))+sinC.cosC
=sin(pi-C).cos(A-B)+sin(pi-A-B).sinC
=sinC.cosA.cosB+sinA.sinB.sinC-sinC.cosA.cosB+sinA.sinB.sinC
=2sinA.sinB.sinC

2)on a pas etudié cotangente

Sujet: Re: A+B+C= pi Dim 25 Jan 2009, 22:13

boomer
cotan =1/tan......

Sujet: Re: A+B+C= pi Dim 25 Jan 2009, 22:20

parfait pour 1)

pour 2) comme a dit Boomer : cotan = 1/tan

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Sujet: Re: A+B+C= pi Dim 25 Jan 2009, 23:29

ok donc
2)sinA/sinBsinC=sin(pi-B-C)/sinBsinC
=(sinBcosC+cosBsinC)/sinBsinC
=cosC/sinC+cosB/sinB
=cotanC+cotanB

sinB/sinAsinC=sin(pi-A-C)/sinAsinC
=(sinAcosC+cosAsinC)/sinAsinC
=cosC/sinC+cosA/sinA
=cotanC+cotanA

sinC/sinAsinB=sin(pi-A-B)/sinAsinB
=(sinAcosB+cosAsinB)/sinAsinB
=cosB/sinB+cosA/sinA
=cotanB+cotanA
donc
sinA/sinBsinC + sinB/sinAsinC + sinC/sinAsinB =

2( cotanA + cotanB + cotanC)

Sujet: Re: A+B+C= pi Lun 26 Jan 2009, 09:58

pour le 1 remarquer seulement que sin(A+B+C)=0 Wink