cf帮帮福利中心:高中数学人教A版必修4第3章《三角恒等变换》章节检测试题(学生版)

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高中数学学习材料
(灿若寒星 精心整理制作)
一、填空题(本大题共 14 小题,每小题 5 分,共 70 分,把答案填在题中横线上) π
1.1-tatna1n221π2=__________. 2.已知 sinθ2+cosθ2=23 3,那么 sinθ=__________,cos2θ=__________. 3.若 sin(α-β)cosα-cos(β-α)sinα=45,则 cos2β=________ 4.设 α∈(0,2π),若 sinα=35,则 2cos(α+π4)等于__________. 5.sin163°sin223°+sin253°sin313°的值为__________. 6.已知 sin(4π-x)=35,则 sin2x 的值为__________. 7.已知 A,B,C 是△ABC 的三个内角,且 tanA,tanB 是方程 3x2-5x+1=0 的两个实数根,则△ABC 是 ________三角形. 8.化简 2 1+sin4+ 2+2cos4的结果是__________. 9.在△ABC 中,若 sin2B=sinAsinC,则 cos2B+cosB+cos(A-C)的值为__________.

14.在△ABC 中,A,B,C 是其三个内角,设 f(B)=4sinB·cos2??π4-B2??+cos2B.当 f(B)-m<2 恒成立时,实
数 m 的取值范围是__________. 二、解答题(本大题共 6 小题,共 90 分,解答时应写出必要的文字说明、证明过程或演算步骤) 15.(本小题满分 14 分)
已知 cos2θ=78,θ∈??2π,π??,求 sin??θ+π6??-sin2θ 的值.
16.(本小题满分 14 分) 已知 tan(π4+θ)+tan(π4-θ)=4,且-π<θ<-2π,求 sin2θ-2sinθcosθ-cos2θ 的值.
17.(本小题满分 14 分) 在△ABC 中,已知 tanB=sinAco+s?sCin-?CB-? B?,试判断△ABC 的形状.

18.(本小题满分 16 分) 求证:2-2sicno?αs4+α-34πs?icno4αs?α+4π?=11+ -ttaannαα.

20.(本小题满分 16 分) 已知 A,B,C 是△ABC 的三个内角,若co1s+2B-sins2inB2B=2+ 3,求角 B.


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