高中數(shù)學(xué)第一章三角函數(shù)1.8函數(shù)的圖像引入輔助角變形及其應(yīng)用素材北師大版4講解

1、1引入輔助角變形及其應(yīng)用引入輔助角變形及其應(yīng)用令22cosaab???，22sin.bab???則22sincossin()axbxabx?????這一變形在三角問題中的應(yīng)用很廣，下面舉例進(jìn)行分類解析，供參考.一、恒等變形1求值例1已知非零實(shí)數(shù)ab滿足sincos855tan15cossin55abab????????，求ba的值.解：????158tan5sin5cos5cos5sin?????baba158tan5sin5cos5c

2、os5sin22222222????????????babbaababbaa引入輔助角?，使22cosbaa???，.sin22bab???則.158tan)5cos()5sin(158tan5sinsin5coscos5cossin5sincos?????????????????????即)158tan()5tan(???????k，?1585???????k，即).(3Zkk??????故?tan?ab.33tan)3tan(???

3、????k注：).(tantanZkk??????????2化簡例2化簡cos()sin()44cos()sin()44??????????????的結(jié)果為（）（A）tan?（B）tan??（C）cos?（D）cos??解：cos()sin()2[cos()]4444cos()sin()2[cos()4444?????????????????????????sintancos???????，選（B）.3證明例3證明：sin50(13ta

4、n10)1.?????3因?yàn)?xx?是函數(shù)()yfx?圖象的一條對稱軸，所以0π26x?πk?，即0π2π6xk??（k?Z）所以0011π()1sin21sin(π)226gxxk?????當(dāng)k為偶數(shù)時(shí)，01π13()1sin12644gx????????????；當(dāng)k為奇數(shù)時(shí)，01π15()1sin12644gx?????（II）1π1()()()1cos21sin2262hxfxgxxx???????????????????1π3

5、1313cos2sin2cos2sin22622222xxxx???????????????????????????1π3sin2232x?????????當(dāng)πππ2π22π232kxk?????，即5ππππ1212kxk????（k?Z）時(shí)，函數(shù)1π3()sin2232hxx?????????是增函數(shù)，故函數(shù)()hx的單調(diào)遞增區(qū)間是5ππππ1212kk????????，（k?Z）3求單調(diào)區(qū)間例7（2007年江蘇文）函數(shù)()sin3

6、cos([0])fxxxx?????的單調(diào)遞增區(qū)間是（）（A）5[]6????（B）5[]66????（C）[0]3??（D）[0]6??解：()sin3cossin()3fxxxx?????，由22()232kxkkZ???????????，解得522().66kxkkZ?????????令0k?，得函數(shù)()sin3cosfxxx??的單調(diào)遞增區(qū)間為[0]6??，選（D）.4求參數(shù)例8（1）要使46sin3cos4mm??????有意