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sin2A=2sinA•cosA cos2A=cos^2 A-sin^2 A=1-2sin^2 A=2cos^2 A-1 tan2A=£¨2tanA£©/£¨1-tan^2 A£©

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sin3¦Á=4sin¦Á¡¤sin(¦Ð/3+¦Á)sin(¦Ð/3-¦Á) cos3¦Á=4cos¦Á¡¤cos(¦Ð/3+¦Á)cos(¦Ð/3-¦Á) tan3a = tan a ¡¤ tan(¦Ð/3+a)¡¤ tan(¦Ð/3-a) Èý±¶½Ç¹«Ê½ÍƵ¼ sin3a =sin(2a+a) =sin2acosa+cos2asina =2sina(1-sin^2a)+(1-2sin^2a)sina =3sina-4sin^3a cos3a =cos(2a+a) =cos2acosa-sin2asina =(2cos^2a-1)cosa-2(1-cos^a)cosa =4cos^3a-3cosa sin3a=3sina-4sin^3a =4sina(3/4-sin^2a) =4sina[(¡Ì3/2)^2-sin^2a] =4sina(sin^260¡ã-sin^2a) =4sina(sin60¡ã+sina)(sin60¡ã-sina) =4sina*2sin[(60+a)/2]cos[(60¡ã-a)/2]*2sin[(60¡ã-a)/2]cos[(60¡ã-a)/2] =4sinasin(60¡ã+a)sin(60¡ã-a) cos3a=4cos^3a-3cosa =4cosa(cos^2a-3/4) =4cosa[cos^2a-(¡Ì3/2)^2] =4cosa(cos^2a-cos^230¡ã) =4cosa(cosa+cos30¡ã)(cosa-cos30¡ã) =4cosa*2cos[(a+30¡ã)/2]cos[(a-30¡ã)/2]*{-2sin[(a+30¡ã)/2]sin[(a-30¡ã)/2]} =-4cosasin(a+30¡ã)sin(a-30¡ã) =-4cosasin[90¡ã-(60¡ã-a)]sin[-90¡ã+(60¡ã+a)] =-4cosacos(60¡ã-a)[-cos(60¡ã+a)] =4cosacos(60¡ã-a)cos(60¡ã+a) ÉÏÊöÁ½Ê½Ïà±È¿ÉµÃ tan3a=tanatan(60¡ã-a)tan(60¡ã+a)

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tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA); cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA. sin^2(a/2)=(1-cos(a))/2 cos^2(a/2)=(1+cos(a))/2 tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))

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sin¦È+sin¦Õ = 2 sin[(¦È+¦Õ)/2] cos[(¦È-¦Õ)/2]

sin¦È-sin¦Õ = 2 cos[(¦È+¦Õ)/2] sin[(¦È-¦Õ)/2] cos¦È+cos¦Õ = 2 cos[(¦È+¦Õ)/2] cos[(¦È-¦Õ)/2] cos¦È-cos¦Õ = -2 sin[(¦È+¦Õ)/2] sin[(¦È-¦Õ)/2] tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB) tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)

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sin¦Ásin¦Â = [cos(¦Á-¦Â)-cos(¦Á+¦Â)] /2 cos¦Ácos¦Â = [cos(¦Á+¦Â)+cos(¦Á-¦Â)]/2 sin¦Ácos¦Â = [sin(¦Á+¦Â)+sin(¦Á-¦Â)]/2 cos¦Ásin¦Â = [sin(¦Á+¦Â)-sin(¦Á-¦Â)]/2

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sinh(a) = [e^a-e^(-a)]/2 cosh(a) = [e^a+e^(-a)]/2 tanh(a) = sin h(a)/cos h(a) ¹«Ê½Ò»£º Éè¦ÁΪÈÎÒâ½Ç,ÖÕ±ßÏàͬµÄ½ÇµÄͬһÈý½Çº¯ÊýµÄÖµÏàµÈ£º sin£¨2k¦Ð£«¦Á£©= sin¦Á cos£¨2k¦Ð£«¦Á£©= cos¦Á tan£¨2k¦Ð£«¦Á£©= tan¦Á cot£¨2k¦Ð£«¦Á£©= cot¦Á ¹«Ê½¶þ£º Éè¦ÁΪÈÎÒâ½Ç,¦Ð+¦ÁµÄÈý½Çº¯ÊýÖµÓë¦ÁµÄÈý½Çº¯ÊýÖµÖ®¼äµÄ¹Øϵ£º sin£¨¦Ð£«¦Á£©= -sin¦Á cos£¨¦Ð£«¦Á£©= -cos¦Á tan£¨¦Ð£«¦Á£©= tan¦Á cot£¨¦Ð£«¦Á£©= cot¦Á ¹«Ê½Èý£º ÈÎÒâ½Ç¦ÁÓë -¦ÁµÄÈý½Çº¯ÊýÖµÖ®¼äµÄ¹Øϵ£º sin£¨-¦Á£©= -sin¦Á cos£¨-¦Á£©= cos¦Á tan£¨-¦Á£©= -tan¦Á cot£¨-¦Á£©= -cot¦Á ¹«Ê½ËÄ£º ÀûÓù«Ê½¶þºÍ¹«Ê½Èý¿ÉÒԵõ½¦Ð-¦ÁÓë¦ÁµÄÈý½Çº¯ÊýÖµÖ®¼äµÄ¹Øϵ£º sin£¨¦Ð-¦Á£©= sin¦Á cos£¨¦Ð-¦Á£©= -cos¦Á tan£¨¦Ð-¦Á£©= -tan¦Á cot£¨¦Ð-¦Á£©= -cot¦Á ¹«Ê½Î壺 ÀûÓù«Ê½-ºÍ¹«Ê½Èý¿ÉÒԵõ½2¦Ð-¦ÁÓë¦ÁµÄÈý½Çº¯ÊýÖµÖ®¼äµÄ¹Øϵ£º sin£¨2¦Ð-¦Á£©= -sin¦Á cos£¨2¦Ð-¦Á£©= cos¦Á tan£¨2¦Ð-¦Á£©= -tan¦Á cot£¨2¦Ð-¦Á£©= -cot¦Á ¹«Ê½Áù£º ¦Ð/2¡À¦Á¼°3¦Ð/2¡À¦ÁÓë¦ÁµÄÈý½Çº¯ÊýÖµÖ®¼äµÄ¹Øϵ£º sin£¨¦Ð/2+¦Á£©= cos¦Á cos£¨¦Ð/2+¦Á£©= -sin¦Á tan£¨¦Ð/2+¦Á£©= -cot¦Á cot£¨¦Ð/2+¦Á£©= -tan¦Á sin£¨¦Ð/2-¦Á£©= cos¦Á cos£¨¦Ð/2-¦Á£©= sin¦Á tan£¨¦Ð/2-¦Á£©= cot¦Á cot£¨¦Ð/2-¦Á£©= tan¦Á sin£¨3¦Ð/2+¦Á£©= -cos¦Á cos£¨3¦Ð/2+¦Á£©= sin¦Á tan£¨3¦Ð/2+¦Á£©= -cot¦Á cot£¨3¦Ð/2+¦Á£©= -tan¦Á sin£¨3¦Ð/2-¦Á£©= -cos¦Á cos£¨3¦Ð/2-¦Á£©= -sin¦Á tan£¨3¦Ð/2-¦Á£©= cot¦Á cot£¨3¦Ð/2-¦Á£©= tan¦Á (ÒÔÉÏk¡ÊZ) A¡¤sin(¦Øt+¦È)+ B¡¤sin(¦Øt+¦Õ) = ¡Ì{(A^2 +B^2 +2ABcos(¦È-¦Õ)} • sin{ ¦Øt + arcsin[ (A•sin¦È+B•sin¦Õ) / ¡Ì{A^2 +B^2; +2ABcos(¦È-¦Õ)} } ¡Ì±íʾ¸ùºÅ,°üÀ¨{¡­¡­}ÖеÄÄÚÈÝ

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sin(-¦Á) = -sin¦Á cos(-¦Á) = cos¦Á tan (-¦Á)=-tan¦Á sin(¦Ð/2-¦Á) = cos¦Á cos(¦Ð/2-¦Á) = sin¦Á sin(¦Ð/2+¦Á) = cos¦Á cos(¦Ð/2+¦Á) = -sin¦Á sin(¦Ð-¦Á) = sin¦Á cos(¦Ð-¦Á) = -cos¦Á sin(¦Ð+¦Á) = -sin¦Á cos(¦Ð+¦Á) = -cos¦Á tanA= sinA/cosA tan£¨¦Ð/2£«¦Á£©£½£­cot¦Á tan£¨¦Ð/2£­¦Á£©£½cot¦Á tan£¨¦Ð£­¦Á£©£½£­tan¦Á tan£¨¦Ð£«¦Á£©£½tan¦Á ÓÕµ¼¹«Ê½¼Ç±³¾÷ÇÏ£ºÆæ±äż²»±ä,·ûºÅ¿´ÏóÏÞ

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sin¦Á=2tan(¦Á/2)/[1+tan²(¦Á/2)] cos¦Á=[1-tan²(¦Á/2)]/[1+tan²(¦Á/2)] tan¦Á=2tan(¦Á/2)/[1-tan²(¦Á/2)]

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(1) (sin¦Á)^2+(cos¦Á)^2=1 (2)1+(tan¦Á)^2=(sec¦Á)^2 (3)1+(cot¦Á)^2=(csc¦Á)^2 Ö¤Ã÷ÏÂÃæÁ½Ê½,Ö»Ð轫һʽ,×óÓÒͬ³ý(sin¦Á)^2,µÚ¶þ¸ö³ý(cos¦Á)^2¼´¿É (4)¶ÔÓÚÈÎÒâ·ÇÖ±½ÇÈý½ÇÐÎ,×ÜÓÐ tanA+tanB+tanC=tanAtanBtanC Ö¤: A+B=¦Ð-C tan(A+B)=tan(¦Ð-C) (tanA+tanB)/(1-tanAtanB)=(tan¦Ð-tanC)/(1+tan¦ÐtanC) ÕûÀí¿ÉµÃ tanA+tanB+tanC=tanAtanBtanC µÃÖ¤ ͬÑù¿ÉÒÔµÃÖ¤,µ±x+y+z=n¦Ð(n¡ÊZ)ʱ,¸Ã¹ØϵʽҲ³ÉÁ¢ ÓÉtanA+tanB+tanC=tanAtanBtanC¿ÉµÃ³öÒÔϽáÂÛ (5)cotAcotB+cotAcotC+cotBcotC=1 (6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2) (7)(cosA£©^2+(cosB£©^2+(cosC£©^2=1-2cosAcosBcosC (8)£¨sinA£©^2+£¨sinB£©^2+£¨sinC£©^2=2+2cosAcosBcosC ÆäËû·ÇÖصãÈý½Çº¯Êý csc(a) = 1/sin(a) sec(a) = 1/cos(a)

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Ìáʾ£º¡Òsin x dx = -cos x + C ¡Ò cos x dx = sin x + C ¡Òtan x dx = ln |sec x | + C ¡Òcot x dx = ln |sin x | + C ¡Òsec x dx = ln |sec x + tan x | + C ¡Òcsc x dx = ln |csc x ¨C cot x | + C ¡Òsin ²x dx =1/2x -1/4 sin 2x + C ¡Ò cos ²x d...

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