2012年解三角形常用变换公式及经典例题.

2012年解三角形常用变换公式及经典例题

两角和公式

sin(A+B) = sinAcosB+cosAsinB sin(A-B) = sinAcosB-sinBcosA  cos(A+B) = cosAcosB-sinAsinB cos(A-B) = cosAcosB+sinAsinB

tan(A+B) = (tanA+tanB)/(1-tanAtanB) tan(A-B) = (tanA-tanB)/(1+tanAtanB) cot(A+B) = (cotAcotB-1)/(cotB+cotA)  cot(A-B) = (cotAcotB+1)/(cotB-cotA) 倍角公式

tan2A = 2tanA/[1-(tanA)^2]

cos2a = (cosa)^2-(sina)^2=2(cosa)^2 -1=1-2(sina)^2 sin2A = 2sinA*cosA 三倍角公式

sin3a = 3sina-4(sina)^3 cos3a = 4(cosa)^3-3cosa

tan3a = tana*tan(π/3+a)*tan(π/3-a) 半角公式

sin(A/2) = √((1-cosA)/2) sin(A/2)=-√((1-cosA)/2) cos(A/2) = √((1+cosA)/2) cos(A/2)=-√((1+cosA)/2)

tan(A/2) = √((1-cosA)/((1+cosA)) tan(A/2)=-√((1-cosA)/((1+cosA)) cot(A/2) = √((1+cosA)/((1-cosA)) cot(A/2)=-√((1+cosA)/((1-cosA))  tan(A/2) = (1-cosA)/sinA=sinA/(1+cosA) 和差化积

sin(a)+sin(b) = 2sin((a+b)/2)cos((a-b)/2) sin(a)-sin(b) = 2cos((a+b)/2)sin((a-b)/2) cos(a)+cos(b) = 2cos((a+b)/2)cos((a-b)/2) cos(a)-cos(b) = -2sin((a+b)/2)sin((a-b)/2) tanA+tanB=sin(A+B)/cosAcosB 积化和差公式

sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)] cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)] sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)] 诱导公式

sin(-a) = -sin(a) cos(-a) = cos(a) sin(pi/2-a) = cos(a) cos(pi/2-a) = sin(a) sin(pi/2+a) = cos(a) cos(pi/2+a) = -sin(a) sin(pi-a) = sin(a) cos(pi-a) = -cos(a) sin(pi+a) = -sin(a) cos(pi+a) = -cos(a) tgA=tanA = sinA/cosA 万能公式

sin(a) = (2tan(a/2))/(1+tan^2(a/2)) cos(a) = (1-tan^2(a/2))/(1+tan^2(a/2)) tan(a) = (2tan(a/2))/(1-tan^2(a/2))