在△ABC中,设AB=c,AC=b,CB=a,s=(a+b+c)/2 , r为内切圆半径, R为外接圆半径,“√”为根号. 1.面积公式S=(1/2)a×ha S=(1/2)ab×sinC S=rs S=abc/(4R) S=2R2×sinAsinBsinC S=s(s-a)×tan(A/2) S=√[(s-a)(s-b)(s-c)s] (海伦公式) S=s2×tan(A/2)tan(B/2)tan(C/2) S=(a2-b2)sinAsinB/[2sin(A-B)] 2.中线.a边中线长Ma=(1/2)×√(2b2+2c2-a2) =(1/2)×√(b2+c2+2bc×cosA) 3.高.a边高长ha=c×sinB=b×sinC ha=a×sinBsinC/sinA ha=√[b2-(a2+b2-c2)2/(2a)2 ] 4.角平分线.a边角平分线长la=2bc×cos(A/2)/(b+c) la=√{bc[(b+c)2-a2]}/(b+c) 5.内切圆,外接圆半径: r=S/s=4R×sin(A/2)sin(B/2)sin(C/2) r=s×tan(A/2)tan(B/2)tan(C/2) R=a/(2sinA)=abc/(4s)=abc/[2r(a+b+c)] 6.同角三角函数间的关系: sinα×cscα=1 cosα×secα=1 tanα×cotα=1 tanα=sinα/cosα,cotα=cosα/sinα (sinα)2+(cosα)2=1 1+(tanα)2=(secα)2 1+(cotα)2=(cscα)2 7.正弦定理: a/sinA=b/sinB=c/sinC=2R 8.余弦定理: a2=b2+c2-2bc cosA b2=a2+c2-2ac cosB c2=a2+b2-2ab cosC 9.倍角公式: sin(2α)=2sinαcosα cos(2α)=(cosα)2-1=1-2(sinα)2 tan(2α)=2tanα/[1-(tanα)2] sin(3α)=3sinα-4(sinα)^3 cos(3α)=4(cosα)^3-3cosα |
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