三角恒等变换公式大全

1、三角函数cos(a+ B)=CoSa'-cosB -sina - sinBcos(a-B)=cos a-cosB +sina - sinBsin(a+ B)=S in a'-cosB +cosa - sinBsin(a-B)=sin a-cosB -cos,a sinBtan(a+ B)=(ta n a+ta nB)/(1-tana - tanB)tan(a-B)=(ta n a-ta nB)/(1+ta na - tanB)二倍角sin (2a) =2sin a - cos a =2tan (a)/1-ta门八2(a)cos (2 a) =cosA2 (a) -si 门八2

2、(a) =2cosA2 (a)-1=1-2si nA2 (a)=1-ta 门八2(a)/1+tanA2(a)tan (2a) =2tan a /1 -ta门八2(a)三倍角sin3 a =3sin a -4sinW (a)C0S3 a =4COSA3 (a)- 3C0S atan3 a = (3tan a -ta门八3 (a)*( 1-3ta门八2 (a)sin3 a =4sin aX sin ( 60- a) sin (60+a)C0S3 a =4cos aX COS ( 60- a) C0s ( 60+a)tan3 a =tan aX tan ( 60- a) tan (60+a)半角公式

3、sin A2(a /2 )=(1-cos a) /2cosA2(a /2 )=(1+cos a) /2tan A2(a /2 )=(1-CoS a) / ( 1+cos a)tan (a /2 ) =sina / ( 1+cos a) = ( 1- CoS a) /si n a半角变形sinA2 (a /2 ) = (1-cosa) /2sin(a/2 ) =V (1-cosa) /2 a/2 在一、二象限=-V (1-cosa) /2 a/2 在三、四象限C0SA2 (a /2 ) = (1+cos a) /2cos(a/2 ) =V (1+cosa) /2 a/2 在一、四象限=-V (1

4、+cosa) /2 a/2 在二、三象限tan A2 (a 12 ) = ( 1-COSa) / ( 1+COSa)tan (a /2 ) =S in a / ( 1+COS a) =( 1- COS a) /si n a =V ( 1-COS a) / ( 1+COS a) a/2在一、三象限=-V ( 1- COS a) / ( 1+COS a) a/2 在二、四象限恒等变形tan(a+ n /4 ) =(tana+1 ) / (1-tana)tan (a- n /4 ) =(tana-1 ) / (1+tana)asinx+bcosx= V( aA2+bA2) a/ V( aA2+bA2

5、) sinx+b/ V(aA2+bA2) cosx= V( aA2+bA2) sin(x+y)(辅助角公式)tan y=b/a万能代换半角的正弦、余弦和正切公式(降幕扩角公式)sin a =2ta n (a /2 ) /1+ta 门八2 (a /2 )cos a =1 -tan (a /2 ) /1+ta 门八2 (a /2 )tan a =2ta n (a /2 ) /1-ta 门八2(a /2 )积和化差sin a cos B :=(1/2 ) sin (a +B ) +si n(a-B)cos a sin B =:(1/2 ) sin (a +B ) -sin(a-B)COS a COS

6、 B =:(1/2 ) cos (a +B ) +COS(a-B)sin a sin B =:-(1/2 ) COS (a + B ) -COS (a - B ）（注：留意最前面是负号）和差化积sin a +sin B =2sin(a + B )/2cos(a -B)/2sin a - sin B =2cos(a + B )/2s in(a -B)/2COS a +COS B =2COS(a + B )/2cos(a -B)/2cos a - cos B=-2sin(a + B )/2si n(a-B)/2内角公式sinA+sinB+sinC=4cos (A/2) cos ( B/2) cos

7、 (C/2) cosA+cosB+cosC=1+4sin (A/2) sin (B/2) sin (C/2 ) tan A+ta nB+ta nC=ta nAta nBta nCcot (A/2) +cot (B/2) +cot (C/2) =cot (A/2) cot ( B/2) cot (C/2)tan( A/2)ta n(B/2)+ta n(B/2)ta n(C/2)+ta n(C/2)ta n(A/2)=1 cotAcotB+cotBcotC+cotCcotA=1证明方法首先，在三角形ABC中,角A,B,C所对边分别为a,b,c若A,B均为锐角，贝恠三 角形ABC中，过C作AB边垂线交AB于D由CD=asinB=bsinA(做另两边的垂线， 同理)可证明正弦定理：a/sinA=b/sinB=c/sinC于是有：AD+BD=cAD=bcosA,BD=acosB AD+BD=代入正弦定理，可得 sinC=sin(180-C)=sin(A+B)=sinAcosB+sinBcosA 即在A,B均为锐角的情况下，可证明正 弦和的公式。