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第19講三角恒等變換(精講)題型目錄一覽①公式的直接應(yīng)用②輔助角公式的應(yīng)用③三角函數(shù)式的化簡(jiǎn)④給值求值問(wèn)題⑤給值求角問(wèn)題一、知識(shí)點(diǎn)梳理一、知識(shí)點(diǎn)梳理一、兩角和與差的正余弦與正切①SKIPIF1<0; ②SKIPIF1<0;③SKIPIF1<0; 二、二倍角公式①SKIPIF1<0;②SKIPIF1<0; ③SKIPIF1<0;三、降冪公式SKIPIF1<0四、輔助角公式SKIPIF1<0(其中SKIPIF1<0).【常用結(jié)論】拆分角的變形:①SKIPIF1<0;SKIPIF1<0;②SKIPIF1<0;③SKIPIF1<0;④SKIPIF1<0;⑤SKIPIF1<0.二、題型分類精講二、題型分類精講題型一公式的直接應(yīng)用策略方法應(yīng)用公式化簡(jiǎn)求值的策略(1)首先要記住公式的結(jié)構(gòu)特征和符號(hào)變化規(guī)律.例如兩角差的余弦公式可簡(jiǎn)化為“同名相乘,符號(hào)相反”.(2)注意與同角三角函數(shù)基本關(guān)系、誘導(dǎo)公式的綜合應(yīng)用.(3)注意配方法、因式分解和整體代換思想的應(yīng)用.【典例1】SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【典例2】下列各式中,值為SKIPIF1<0的是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【題型訓(xùn)練】一、單選題1.SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<02.SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<03.設(shè)SKIPIF1<0,則SKIPIF1<0等于(
)A.-2 B.2 C.-4 D.44.已知角SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<05.若SKIPIF1<0為銳角,SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.1 C.SKIPIF1<0 D.SKIPIF1<06.已知SKIPIF1<0,則SKIPIF1<0的值是(
)A.SKIPIF1<0 B.2 C.SKIPIF1<0 D.SKIPIF1<0二、填空題7.計(jì)算:SKIPIF1<0______.8.若cosα=-SKIPIF1<0,α是第三象限的角,則sinSKIPIF1<0=____________.9.SKIPIF1<0____________.10.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0__________.題型二輔助角公式的應(yīng)用【典例1】求函數(shù)SKIPIF1<0的最大值(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【題型訓(xùn)練】一、單選題1.該函數(shù)SKIPIF1<0的最大值是(
)A.1 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<02.函數(shù)SKIPIF1<0的最小正周期和最大值分別是(
)A.SKIPIF1<0和SKIPIF1<0 B.SKIPIF1<0和2 C.SKIPIF1<0和SKIPIF1<0 D.SKIPIF1<0和23.)已知SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<04.SKIPIF1<0的值所在的范圍是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0二、填空題5.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的最大值為________6.若SKIPIF1<0,則SKIPIF1<0__________.7.函數(shù)SKIPIF1<0的最大值為______.題型三三角函數(shù)式的化簡(jiǎn)策略方法1.三角函數(shù)式的化簡(jiǎn)要遵循“三看”原則2.三角函數(shù)式化簡(jiǎn)的方法(1)弦切互化,異名化同名,異角化同角,降冪或升冪.(2)在三角函數(shù)式的化簡(jiǎn)中“次降角升”和“次升角降”是基本的規(guī)律,根號(hào)中含有三角函數(shù)式時(shí),一般需要升次.【典例1】已知SKIPIF1<0,則SKIPIF1<0的值是(
)A.SKIPIF1<0 B.不存在 C.SKIPIF1<0或不存在 D.SKIPIF1<0【典例2】已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【題型訓(xùn)練】一、單選題1.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<02.已知SKIPIF1<0,則SKIPIF1<0等于(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<03.已知銳角SKIPIF1<0,SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0的值為(
)A.1 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<04.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<05.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<06.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<07.已知SKIPIF1<0,則SKIPIF1<0的值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0二、填空題8.函數(shù)SKIPIF1<0的最小正周期為__________.9.已知SKIPIF1<0為鈍角,SKIPIF1<0,則SKIPIF1<0的值為______.10.已知SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0的值是______.題型四給值求值問(wèn)題策略方法給值求值:一般是給出某些角的三角函數(shù)值,求另外一些角的三角函數(shù)值,解題的關(guān)鍵在于“變角”,使相關(guān)角相同或具有某種關(guān)系.【典例1】已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【題型訓(xùn)練】一、單選題1.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的值為(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<02.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<03.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.1 D.SKIPIF1<04.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<05.已知SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<06.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<07.已知SKIPIF1<0,則SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<08.已知SKIPIF1<0,SKIPIF1<0為銳角,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<09.已知SKIPIF1<0,且SKIPIF1<0,則cosβ=(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.0二、填空題10.已知SKIPIF1<0,則SKIPIF1<0__________.11.已知SKIPIF1<0,則SKIPIF1<0________.12.已知SKIPIF1<0,則SKIPIF1<0__________.13.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0________.題型五給值求角問(wèn)題策略方法給值求角:實(shí)質(zhì)上可轉(zhuǎn)化為“給值求值”,即通過(guò)求角的某一個(gè)三角函數(shù)值來(lái)求角.在選取函數(shù)時(shí),遵循以下原則:①已知正切函數(shù)值,選正切函數(shù).②已知正、余弦函數(shù)值,若角的范圍是0,π2,選正、余弦函數(shù)皆可,若角的范圍是(0,π),選余弦函數(shù),若角的范圍是【典例1】已知SKIPIF1<0,則SKIPIF1<0的值可能為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0【題型訓(xùn)練】一、單選題1.設(shè)SKIPIF1<0,SKIPIF1<0是方程SKIPIF1<0的兩根,且SKIPIF1<0,則SKIPIF1<0(
).A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0或SKIPIF1<0 D.SKIPIF1<02.已知SKIPIF1<0且SKIPIF1<0,則SKIPIF1<0=(
)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0或SKIPIF1<03.已知SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0的值是(
)A.
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