2024年数学高考大一轮复习第四章 §4.4　简单的三角恒等变换

§4.4　简单的三角恒等变换

考试要求　能运用两角和与差的正弦、余弦、正切公式推导二倍角的正弦、余弦、正切公式，并进行简单的恒等变换(包括推导出积化和差、和差化积、半角公式，这三组公式不要求记忆)．

知识梳理

1．二倍角的正弦、余弦、正切公式

(1)公式S2α：sin 2α＝____________________.

(2)公式C2α：cos 2α＝_______________＝_______________＝______________.

(3)公式T2α：tan 2α＝____________________.

2．常用的部分三角公式

(1)1－cos α＝________________，1＋cos α＝________________.(升幂公式)

(2)1±sin α＝__________________________.(升幂公式)

(3)sin2α＝________________，cos2α＝__________________，tan2α＝________.(降幂公式)

思考辨析

判断下列结论是否正确(请在括号中打“√”或“×”)

(1)半角的正弦、余弦公式实质就是将倍角的余弦公式逆求而得来的．(　　)

(2)存在实数α，使tan 2α＝2tan α.(　　)

(3)cos2＝.(　　)

(4)tan ＝＝.(　　)

教材改编题

1．(2021·全国乙卷)cos2－cos2等于(　　)

A.  B.  C.  D.

2．若角α满足sin α＋2cos α＝0，则tan 2α等于(　　)

A．－  B.  C．－  D.

3．化简的结果是(　　)

A.cos 10°   B．－cos 10°

C.sin 10°   D．－sin 10°

题型一　三角函数式的化简

例1 (1)(2021·全国甲卷)若α∈，tan 2α＝，则tan α等于(　　)

A.  B.  C.  D.

(2)已知sin α＋cos α＝，则sin2＝________.

听课记录：______________________________________________________________

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思维升华 (1)三角函数式的化简要遵循“三看”原则：

一看角，二看名，三看式子结构与特征．

(2)三角函数式的化简要注意观察条件中角之间的联系(和、差、倍、互余、互补等)，寻找式子和三角函数公式之间的联系点．

跟踪训练1 (1)若f(α)＝2tan α－，则f 的值是________．

(2)已知0<θ<π，则＝________.

题型二　三角函数式的求值

命题点1　给角求值

例2 计算：(1)sin 10°·sin 30°·sin 50°·sin 70°；

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(2)－；

(3).

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命题点2　给值求值

例3 (2023·长春质检)已知sin＋cos α＝，则sin等于(　　)

A.  B.  C．－  D．－

听课记录：______________________________________________________________

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命题点3　给值求角

例4 已知 sin α＝，cos β＝，且α，β为锐角，则α＋2β＝________.

听课记录：______________________________________________________________

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思维升华 (1)给值(角)求值问题求解的关键在于“变角”，使其角相同或具有某种关系，借助角之间的联系寻找转化方法．

(2)给值(角)求值问题的一般步骤

①化简条件式子或待求式子；

②观察条件与所求式子之间的联系，从函数名称及角入手；

③将已知条件代入所求式子，化简求值．

跟踪训练2 (1)已知α∈(0，π)，sin 2α＋cos 2α＝cos α－1，则sin 2α等于(　　)

A.   B．－

C．－ 或0   D.

(2)(2023·南京模拟)已知sin＝tan 210°，则sin(60°＋α)的值为(　　)

A.  B．－  C.  D．－

题型三　三角恒等变换的综合应用

例5 已知f(x)＝sin＋2sin·cos.

(1)求f 的值；

(2)若锐角α满足f(α)＝，求sin 2α的值．

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思维升华 (1)进行三角恒等变换要抓住：变角、变函数名称、变结构，尤其是角之间的关系；注意公式的逆用和变形使用．

(2)形如y＝asin x＋bcos x化为y＝sin(x＋φ)，可进一步研究函数的周期性、单调性、最值与对称性．

跟踪训练3 已知3sin α＝2sin2－1.

(1)求sin 2α＋cos 2α的值；

(2)已知α∈(0，π)，β∈，2tan2β－tan β－1＝0，求α＋β的值．

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