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1、常见的三角恒等式及其证明设A,B,C是三角形的三个内角 (1) tanAtanBtanC=tanAtanBtanC 证明: tanA+tanB+tanC=tan(A+B)(1-tanAtanB)+tanC=tan(-c)(1-tanAtanB)+tanC=-tanC(1-tanAtanB)+tanC=tanAtanBtanC (2) cotAcotBcotBcotC+cotCcotA=1 证明: tanAtanBtanC=tanAtanBtanC cotX*tanX=1 tanA*cotAcotBcotCtanB*cotAcotBcotCtanC*cotAcotBcotC=tanAtanBta
2、nC*cotAcotBcotC cotAcotBcotBcotC+cotCcotA=1 (3) (cosA)2+(cosB)2+(cosC)2+2cosAcosBcosC=1 证明: (cosA)2+(cosB)2+x2+2cosAcosBx=1 x2+2cosAcosBx+(cosA)2+(cosB)2-1=0 x=-2cosAcosB+-(2cosAcosB)2-4(cosA)2+(cosB)2-1)/2 x=-cosAcosB+-(cosAcosB)2-(cosA)2+(cosB)2-1) x=-cosAcosB+-1-(cosA)21-(cosB)2 x=-cosAcosB+-(sin
3、A)2(sinB)2 x=-cosAcosB+-sinAsinB x=-cos(A+B)或x=-cos(A-B) x=cosC或x=-cos(A-B) 所以 cosC是方程的一个根 所以 (cosA)2+(cosB)2+(cosC)2+2cosAcosBcosC=1 (4) cosA+cosB+cosC14sin(A/2)sin(B/2)sin(C/2) 证明: cosA+cosB+cosC=1+4sin(A/2)sin(B/2)sin(C/2) cos(180-B-C)+cosB+cosC=1+2sin(A/2)2sin(B/2)sin(C/2) cos(180-B-C)+cosB+cosC
4、=1+2cos(B/2+C/2)2sin(B/2)sin(C/2) -cos(B+C)+cosB+cosC=1+2cos(B/2+C/2)2sin(B/2)sin(C/2) -cos(B+C)+cosB+cosC=1+2cos(B/2+C/2)cos(B/2-C/2)-cos(B/2+C/2) -cos(B+C)+cosB+cosC=1+2cos(B/2+C/2)cos(B/2-C/2)-2cos(B/2+C/2)2 cosB+cosC=2cos(B/2+C/2)cos(B/2-C/2) 2cos(B/2+C/2)2-1=cos(B+C) (5) tan(A/2)tan(B/2)+tan(B/
5、2)tan(C/2)+tan(C/2)tan(A/2)=1 证明: A/2+B/2+C/2=/2 (/2-A)+(/2-B)+(/2-C)= cot(/2-A)cot(/2-B)+cot(/2-C)cot(/2-B)+cot(/2-A)cot(/2-C)=1 tan(A/2)tan(B/2)+tan(B/2)tan(C/2)+tan(C/2)tan(A/2)=1 (6) sin2A+sin2B+sin2C=4sinAsinBsinC 证明: 设三角形ABC的外心为O SABO+SACO+SCBO=SABC (1/2)RRsin2C+(1/2)RRsin2B+(1/2)RRsin2A=(1/2)
6、bcsinA=(1/2)2RsinB*2RsinC*sinA sin2A+sin2B+sin2C=4sinAsinBsinC (7) sinA+sinB+sinC=4cos(A/2)cos(B/2)cos(C/2) 证明: 4cos(A/2)cos(B/2)cos(C/2) =2cos(C/2)*2cos(A/2)cos(B/2) =2sin(A/2+B/2)*cos(A/2+B/2)+cos(A/2-B/2) =2sin(A/2+B/2)cos(A/2+B/2)+2sin(A/2+B/2)cos(A/2-B/2) =sin(A+B)+2sin(A/2+B/2)cos(A/2-B/2) =sinC+2sin(A+B)/2cos(A-B)/2 =sinC+sin(A+B)/2+(A-B)/2+sin(A+B)/2-(A-B)/2 =sinC+sinA+sinB