Question

数学三角恒等变换常用的推论

全一点

Answer 1

sin(-a) = -sina
cos(-a) = cosa
sin(π/2-a) = cosa
cos(π/2-a) = sina
sin(π/2+a) = cosa
cos(π/2+a) = -sina
sin(π-a) = sina
cos(π-a) = -cosa
sin(π+a) = -sina
cos(π+a) = -cosa
两角和公式
sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB-cosAsinB
cos(A+B) = cosAcosB-sinAsinB
cos(A-B) = cosAcosB+sinAsinB
tan(A+B) = (tanA+tanB)/(1-tanAtanB)
tan(A-B) = (tanA-tanB)/(1+tanAtanB)
倍角公式
tan2A = 2tanA/(1-tan²A)
sin2A=2SinA•CosA
cos2A = Cos2A-Sin2A=2Cos2A-1=1-2sin2A
万能公式。（这个可以巧解很多题）
设tan(A/2)=t
sinA=2t/(1+t^2)
tanA=2t/(1-t^2)
cosA=(1-t^2)/(1+t^2)

推导第一个： (其它类似)
sinA=2sin(A/2)cos(A/2)
=[2sin(A/2)cos(A/2)]/[sin^2(A/2)+cos^2(A/2)]
分子分母同时除以cos^2(A/2)
=[2sin(A/2)cos(A/2)/cos^2(A/2)]/[(sin^2(A/2)+cos^2(A/2))/cos^2(A/2)]
化简：
=[2sin(A/2)/cos(A/2)]/[sin^2(A/2)/cos^2(A/2)+1]
即：
=(2tan(A/2))/(tan^(A/2)+1)

Answer 2

sin(-a) = -sina
cos(-a) = cosa
sin(π/2-a) = cosa
cos(π/2-a) = sina
sin(π/2+a) = cosa
cos(π/2+a) = -sina
sin(π-a) = sina
cos(π-a) = -cosa
sin(π+a) = -sina
cos(π+a) = -cosa
两角和公式
sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB-cosAsinB
cos(A+B) = cosAcosB-sinAsinB
cos(A-B) = cosAcosB+sinAsinB
tan(A+B) = (tanA+tanB)/(1-tanAtanB)
tan(A-B) = (tanA-tanB)/(1+tanAtanB)
倍角公式
tan2A = 2tanA/(1-tan²A)
sin2A=2SinA•CosA
cos2A = Cos2A-Sin2A=2Cos2A-1=1-2sin2A
积化和差公式现在高中不要求掌握了。