三角函数公式及其推导1.三角函数的定义由此,我们定义:如FigureI,在ΔABC中sin()cos()tan()11cot()tan11sec()cos11csc()sinbcacbaabbacaaccbbc对边的正弦值:斜边邻边的余弦值:斜边对边的正切值:邻边邻边的余切值:对边斜边的正割值:邻边斜边的余割值:对边备注:当用一个字母或希腊字母表示角时,可略写∠符号,但用三个子母表示时,不能省略。在本文中,我们只研究sin、cos、tan。2.额外的定义222222sin(sin)cos(cos)tan(tan)AcbθCaBFigureI3.简便计算公式22sincoscos(90)cossinsin(90)111tantantan(90)sincos1bAccAbbaaAb证明:2222222222901sinsin1sincos1ABCABCabcabccBA在中,证完222222sintancossincos1tan1coscoscosbbcaac4.任意三角形的面积公式如FigureII,CabhdeBcAFigureII121sin21sin()2ABCSahabCacB两边和其夹角正弦的乘积5.余弦定理:任意三角形一角的余弦等于两邻边的平方和减对边的平方之差与两邻边积的两倍之比。证明:如FigureII,2222222222222222222222(cos)(sin)2coscossin=2cos(cossin)2coscos22bdhacBcBaacBcBcBaacBcBBacacBbacacbBacac证完6.海伦公式证明:如FigureII,2222244422222222224442222222222444222222222244422221sin211cos21122122212414222241422244214ABCSabCabCabcabababcabacbcababababcabacbcababababcabacbcabababcabab22222222416acbcababcabcbcaabc222222222222222222=2ABCabccabcbabcaabcabccabcbabcaabcabcabcabcabcabcabcsSssasbsc设:7.正弦定理如FigureIII,c为ΔABC外接圆的直径,sin2sinaAcacrrABCA(为的外接圆半径)同理:,sinsin2sinsinsinbcccBCabcrABCAcOBaCFigureIII8.加法定理(1)两角差的余弦如FigureIV,AOCBOCAOB令AO=BO=r点A的横坐标为cosAxr点A的纵坐标为sinAyr点B的横坐标为cosBxr点B的纵坐标为sinByr22222222222222222222222222sinsincoscossinsin2sinsincoscos2coscossinsin2sinsincoscos2coscossincossincos2sinsin2coscos112sABABAByyxxrrrrrrrrrrrrr22insincoscos22sinsincoscos21sinsincoscosrryABOCxβ(α-β)αFigureIV由余弦公式可得:2222222222cos2cos22cos22cos21cosABACBCACBCACBrrrrrrrr综上得:cossinsincoscos(2)两角和的余弦coscossinsincoscossinsincoscoscoscossinsin(3)两角和的正弦sincos90cos90sin90sincos90coscossinsincos(4)两角差的正弦sinsincossinsincoscossinsincossincoscossin(5)两角和的正切sintancoscossinsincoscoscossinsincossinsincoscoscoscoscossinsincoscossinsincoscossinsin1coscostantan1tantan(6)两角差的正切tantantantan1tantantantan1tantan9.两倍角公式222222222222sin2sinsincossincos2sincoscos2coscoscossinsincossin12sin2cos1sin2tan2cos22sincoscossin2sincoscoscossincos2sincossin1cos2tan1ta2n10.积化和差公式1sincos2sincos21sincossincoscossincossin21sinsin21coscos2coscos21coscoscoscossinsinsinsin21coscos21sinsin2sinsin21sinsinsinsincoscoscoscos21coscos211.和差化积公式(1)设:A=α+β,B=α-β,sinsinsinsinsincoscossinsincoscossin2sincos2sincos222sincos22sinsinsinsinsincoscossinsincoscossin2cossiABABABABn2cossin222cossin22ABAB(2)设:2222cos,sin,ababab∵22cossin1222222222222sinsincoscossinsincossinsinbabababababababa12.其他常用公式000sin360sincos360costan360tansin90coscos90sin1tan90tansin90coscos90sin1tan90tansin90coscos90sin1tan90tansin180sincos180cosnnntan180tansin180sincos180costan180tansinsincoscostantantan21901cos1cos11sin1sin1n不存在13.特殊的三角函数值00151230645460357512902sin06241222326241cos16243222126240tan023331323N/A14.关于机器算法在计算机中,三角函数的算法是这样的,其中x用弧度计算135721002460sin1!3!5!7!21!cos0!2!4!6!2!nnnnxxxxxxnxxxxxxn推导公式:(a+b+c)/(sinA+sinB+sinC)=2R(其中,R为外接圆半径)由正弦定理有a/sinA=b/sinB=c/sinC=2R所以a=2R*sinAb=2R*sinBc=2R*sinC加起来a+b+c=2R*(sinA+sinB+sinC)带入(a+b+c)/(sinA+sinB+sinC)=2R*(sinA+sinB+sinC)/(sinA+sinB+sinC)=2R两角和公式sin(A+B)=sinAcosB+cosAsinBsin(A-B)=sinAcosB-cosAsinBcos(A+B)=cosAcosB-sinAsinBcos(A-B)=cosAcosB+sinAsinBtan(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Sin2A=2SinA?CosA对数的性质及推导用^表示乘方,用log(a)(b)表示以a为底,b的对数*表示乘号,/表示除号定义式:若a^n=b(a0且a≠1)则n=log(a)(b)基本性质:1.a^(log(a)(b))=b2.log(a)(MN)=log(a)(M)+log(a)(N);3.log(a)(M/N)=log(a)(M)-log(a)(N);4.log(a)(M^n)=nlog(a)(M)推导1.这个就不用推了吧,直接由定义式可得(把定义式中的[n=log(a)(b)]带入a^n=b)2.MN=M*N由基本性质1(换掉M和N)a^[log(a)(MN)]=a^[log(a)(M)]*a^[log(a)(N)]由指数的性质a^[log(a)(MN)]=a^{[log(a)(M)]+[log(a)(N)]}又因为指数函数是单调函数,所以log(a)(MN)=log(a)(M)+log(a)(N)3.与2类似处理MN=M/N由基本性质1(换掉M和N)a^[log(a)(M/N)]=a^[log(a)(M)]/a^[log(a)(N)]由指数的性质a^[log(a)(M/N)]=a^{[log(a)(M)]-[log(a)(N)]}又因为指数函数