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高三数学一轮复习精品课件两角和与差的三角函数_图文

第3课时 两角和与差的三角函数 基础知识梳理 1.两角和与差的正弦、余弦、正切公式 C(α-β):cos(α-β)= cosαcosβ+sinαsinβ ; C(α+β):cos(α+β)= cosαcosβ-sinαsinβ ; S(α+β):sin(α+β)= sinαcosβ+cosαsinβ ; S(α-β):sin(α-β)=sinαcosβ-cosαsinβ; tanα+tanβ T(α+β):tan(α+β)= 1-tanαtanβ ; tanα-tanβ T(α-β):tan(α-β)= . 1+tanαtanβ 基础知识梳理 2.二倍角的正弦、余弦、正切公式 S2α:sin2α= 2sinαcosα ; C2α:cos2α=cos2α-sin2α= 1-2sin2α = 2cos2α-1 ; 2tanα 2 T2α:tan2α= 1-tan α . 三基能力强化 1.(2009年高考全国卷Ⅰ改编) 已知tanα=4,tanβ=3,则tan(α+ β)=( ) 7 A. 11 7 C. 13 7 B.- 11 7 D.- 13 答案:B 三基能力强化 3 π 2.已知 sinα= ,且 α∈ ( ,π),那 5 2 sin2α 么 2 的值等于( ) cos α 3 3 A.- B.- 4 2 3 3 C. D. 4 2 答案:B 三基能力强化 3.sin163°sin223°+sin253°sin313° =( ) 1 A.- 2 3 C.- 2 1 B. 2 3 D. 2 答案:B 三基能力强化 3 4. (教材习题改编)已知 sinα= , cosβ 5 4 π 3π =- ,α∈ ( ,π),β∈ (π, ),则 sin(α 5 2 2 -β)=________. 24 答案:- 25 三基能力强化 1 π 5.若 cosα= ,其中 α∈ (- ,0), 2 2 α 则 sin 的值是________. 2 1 答案:- 2 课堂互动讲练 考点一 给角求值问题 一般所给出的角都是非特殊角, 从表面来看是很难的,但仔细观察非 特殊角与特殊角总有一定的关系,解 题时,要利用观察得到的关系,结合 三角公式转化为特殊角并且消除非特 殊角的三角函数而得解.有时还可逆 用、变形运用公式. 课堂互动讲练 例1 求值: sin50° (1+ 3tan10° )-cos20° . cos80° 1-cos20° 课堂互动讲练 【思路点拨】 首先把切函数化 为弦函数,同时把根号去掉. 【 解 】 ∵sin50° (1 + 3 tan10° ) cos10° + 3sin10° = sin50° · cos10° 2sin40° = sin50° · =1, cos10° 课堂互动讲练 cos80° 1- cos20° = sin10° 2sin210° = 2sin210° . sin50° (1+ 3tan10° )- cos20° ∴ = cos80° 1- cos20° 1- cos20° = 2. 2 2sin 10° 【名师点评】 在利用二倍角的 余弦时,注意选用公式. 课堂互动讲练 考点二 给值求值问题 三角函数的给值求值问题解决的 关键在于把“所求角”用“已知角”表 示. (1)当“已知角”有两个时,“所求 角”一般表示为两个“已知角”的和或 差的形式. (2)当“已知角”有一个时,此时应 着眼于“所求角”与“已知角”的和或差 的关系,然后应用诱导公式把“所求 角”变成“已知角”. 课堂互动讲练 (3)常见的配角技巧 α α=2· ; 2 α=(α+β)-β; α=β-(β-α); 1 α= [(α+β)+(α-β)]; 2 1 β= [(α+β)-(α-β)]; 2 π π π +α= -( -α). 4 2 4 课堂互动讲练 例2 π 3 π 已知 0<β< <α< π,cos( -α) 4 4 4 3 3π 5 = ,sin( +β)= ,求 sin(α+β) 5 4 13 的值. 课堂互动讲练 【思路点拨】 比较题设中的角与 3π π 待求式中的角,不难发现 ( + β)- ( - 4 4 π π α) = + (α + β) 或 将 cos( - α) 变 化 为 2 4 π π 3π sin( + α), 再由( + α)+ ( +β)= π+ (α 4 4 4 + β)求解. 课堂互动讲练 π 3π 【解】 ∵ <α< , 4 4 3π π π π ∴- <- α<- ,- < - α<0. 4 4 2 4 π 3 π 4 又∵cos( - α)= ,∴sin( - α)=- . 4 5 4 5 π 3π 3π ∵0<β< ,∴ < + β<π. 4 4 4 3π 5 又∵sin( + β)= , 4 13 3π 12 ∴cos( +β)=- , 4 13 课堂互动讲练 π ∴sin(α+ β)=- cos[ + (α+ β)] 2 3π π =- cos[( +β)- ( - α)] 4 4 3π π 3π =- cos( + β)cos( - α)- sin( + 4 4 4 π β)sin( - α) 4 12 3 5 4 =-(- )× - ×(- ) 13 5 13 5 36 20 56 = + = . 65 65 65 课堂互动讲练 误区警示】 在做题时,有时忽略求 π 3π - α, + β 的范围,还有不能正确判断 4 4 两角的范围. 课堂互动讲练 互动探究 在例2条件不变的情况下,求 cos(α-β)的值. π 3π 解:∵ <α< , 4 4 3π π π π ∴- <- α<- ,- < - α<0. 4 4 2 4 π 3 π 又 ∵cos( - α)= , ∴sin( - α) 4 5 4 4 =- . 5 课堂互动讲练 π 3π 3π ∵0<β< ,∴ <

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