三角恒等变换公式大全

三角函数

cos (a+ B)=CoS a'-cos B - sin a - sin B

cos (a-B)=cos a-cos B

+ sin a - sin B

sin (a+ B)=S in a'-cos B cos a - sin B

sin (a-B)=sin a-cos B - cos ,a・sin B

tan (a+ B)=(ta n a+ta n B)/ (1-tan a - tan B)

tan (a-B)=(ta n a-ta n B)/ (1+ta n a - tan B)

二

倍

角

sin (2a) =2sin a - cos a =2tan (a) /[1-ta门(a)]

cos (2 a) =cosA2 (a) -si 门八2 (a) =2cosA2 (a)-1=1-2si nA2 (a)=[1-ta 门

八(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 =4COS A3 (a) - 3C0S a

tan3 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

sin A2 (a /2 )= (1-cos a) /2

cosA2 (a /2 )= (1+cos a) /2

tan A2 (a /2 )= (1-CoS a) / ( 1+cos a)

tan ( a /2 ) =sin a / ( 1+cos a) = ( 1- CoS a) /si n a

半角变形

sinA2 (a /2 ) = (1-cos a) /2

sin(a/2 ) =V[ (1-cos a) /2] a/2 在一、二象限

=-V[ (1-cos a) /2] a/2 在三、四象限

C0SA2 (a /2 ) = (1+cos a) /2

cos(a/2 ) =V[ (1+cos a) /2] a/2 在一、四象限

=-V[ (1+cos a) /2] a/2 在二、三象限

tan A2 (a 12 ) = ( 1-COS a) / ( 1+COS a)

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) ]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 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 ) /2]cos[ (a - B ) /2]

sin a - sin B =2cos[ (a + B ) /2]s in[ (a - B ) /2] COS a +COS B =2COS[

(a + B ) /2]cos[ (a - B ) /2] cos a - cos B=-2sin[ (a + B )

/2]si n[

(a -B) /2]

内角公式

sinA+sinB+sinC=4cos (A/2) COS ( B/2) COS (C/2) cosA+cosB+cosC=1+4sin (A/2) sin (B/2) sin (C/2 ) tan A+ta nB+ta nC=ta nAta nBta nC cot (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，