www.5129.net > 在三角形ABC中,内角ABC所对的边分别为ABC,且满足A+√2/2C=Bsin(A+π/4)

在三角形ABC中,内角ABC所对的边分别为ABC,且满足A+√2/2C=Bsin(A+π/4)

由正弦定理得sinA+(√2/2)sinC=sinBsin(A+π/4)sinA+(√2/2)sin(A+B)=sinB(sinAcosπ/4+cosAsinπ/4)(√2/2)sinAsinB-(√2/2)sinAcosB=sinA(√2/2)sinB-(√2/2)cosB=1sin(B- π/4)=1B=π

2asin(C+π/6) = b+c根据正弦定理有:2sinAsin(C+π/6) = sinB+sinC2sinA{sinCcosπ/6+cosCsinπ/6) = sinB+sinCsinA{√3sinC+cosC) = sinB+sinC√3sinAsinC+sinAcosC = sinB+sinC又,sinB=sin(A+C) = sinAcosC+cosAsinC∴ √3sinAsinC+

在△abc中,角a,b,c所对的边分别为a,b,c,且满足cos(a/2)=(2√5)/5,向量ab乘以向量ac=3 (1)求△abc的面积(2)若b+c=6,求a的值【解】cosa=2(cosa/2)^2-1=2*4/5-1=3/5再根据向量可得ab*ac*cosa=3所以ab*ac=5由cosa>0,在三角形中可以知道∠a为锐角,所以sina=4/5所以面积=1/2ab*ac*sina=2ab*ac=5,即bc=5.a^2 = b^2 + c^2 - 2bc cosa =(b+c)^2 - 2bc -2bc cosa =36 - 10 -10x3/5 =20 a = 2倍根5

1证明:a/sinA=b/sinB=c/sinC.代入题中2(siaA+sinC)=(√3+1)sinB即2*2sin(A/2+C/2)cos(A/2-C/2)=(√3+1)*2sinB/2cosB/2因为A+B+C=180.所以2sin(A/2+C/2)=2sin(90-B/2)=2cosB/2所以有2cos(A/2-C/2)=(√3+1)cosB/2.得证.2若A=2C.代入1中,且由A+B+C=3C+B=180.有C=30,A=60,B=90.3若A+c=90,代入1中且A+B+C=180.有A=60,C=30,或A=30,C=60.B=90.

cosA=-√2/4,则:sinA=√14/4a/sinA=c/sinC,2/[√14/4]=[√2]/[sinC]sinC=√7/4cosC=3/4cosB=-cos(A+C)=(10√2)/(16)=(5√2)/8则:b=a+c-2accosB=4+2-5=1则:b=1

1) a=2bsinA 由正弦定理得:sinA=2sinBsinA sinA≠0,1=2sinB sinB=1/2 ∴B=30° 2)余弦定理有:b 2;=a 2;+c 2;-2accosB=(3√3) 2

余弦定理:c=a+b-2abcosC=4+16-2*2*4*1/2=12,c=2√3.

1.三角形的面积=(bc*sinA)/2.因为:向量AB*向量AC=3,所以:b*c*cosA=3.且:cos(A/2)=2√5/5cosA=2cos^2(A/2)-1=15/25=3/5.所以:sinA=4/5.代人b*c*cosA=3中解得:bc=5.所以:三角形的面积=(bc*sinA)/2=(5*4/5)/2=2.2.因为: 展开 作业帮用户 2016-11-21 举报

解:∵cosA=-√2/4 ∴sinA=√14/4 由正弦定理,有 a/sinA=c/sinC 则 sinC=c*sinA/a =√2*(√14/4)÷2 =√7/4 cosC=3/4 ∵sinB=sin[π-(A+C)]=sin(A+C) ∴sinB=sinA*cosC+cosA*sinC =(√14/4)*(3/4)+(-√2/4)*(√7/4) =√14/8 故 b=a*sinB/sinA =2*

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