伤城文章网 >  > 2014届高考理科理数学第一轮知识点总复习测试题17

2014届高考理科理数学第一轮知识点总复习测试题17






两角和与差的正弦、

余弦和正切公式

【选题明细表】 知识点、方法 三角公式的简单应用 三角公式的逆用与变形用 综合问题 题号 2、3、6、7、9、11 1、4、8 5、10

一、选择题 1.化简 cos 15° cos 45° -cos 75° sin 45° 的值为( A ) (A) (B) (C)- (D)-

解析:cos 15° cos 45° -cos 75° sin 45° = cos 15° cos 45° -sin 15° sin 45° = cos(15° +45° )=cos 60° . = 故选 A. 2.(2012 青岛模拟)已知 tan (A) (B)- (C) (D)= =3, =3,则 tan α的值为( A )

解析:法一 ∵tan

∴tan α= . 法二 tan α=tan = =. 故选 A. 3.(2013 龙岩质检)已知 cos 是( C ) (A)- (B) (C)- (D) 解析: cos 以 sin +sin α= =-sin
? sin α+ cos α=

=

+sin α= ,则 sin

的值

? sin

= ,所

=- .故选 C. ,则 cos(α-β)的值

4.已知 cos α= ,cos(α+β)=- ,且α、 β∈ 等于( D ) (A)- (B) (C)- (D) ,∴α+β∈(0,π), = = ,

解析:∵α、β∈ ∴sin α=

sin(α+β)=

=

= .

∴cos β=cos[(α+β)-α]= cos(α+β)cos α+sin(α+β)sin α= ×+ × =, ∴sin β= = = ,

∴cos(α-β)=cos αcos β+sin αsin β= × + × = .故选 D. 5.已知向量 a= 等于( B ) (A)- (B)- (C) (D) +4cos α- =0, ,b=(4,4cos α- ),若 a⊥b,则 sin

解析:由 a⊥b 得 a·b=0,即 4sin 于是 sin +cos α= ,

因此 sin α+ cos α+cos α= , 即 sin α+ cos α= , 故 sin 于是 sin = ,所以 sin =-sin =, =- .故选 B.

6.在△ABC 中,C=120° ,tan A+tan B= (A) (B) (C) (D)

,则 tan Atan B 的值为( B )

解析:由 C=120° A+B=60° 得 , 于是 tan(A+B)= 即 = ,

= ,所以 tan Atan B= .故选 B.

二、填空题 7. (2012 福州质检)如图所示,点 B 在以 PA 为直径的圆周上,点 C 在 线段 AB 上,已知 PA=5,PB=3,PC= 为锐角,则角β的值为 . ,设∠APB=α,∠APC=β,α、β均

解析:因为点 B 在以 PA 为直径的圆周上, 所以∠ABP=90° , 所以 cos α= = ,sin α= , 即 tan α= , 因为 cos∠CPB=cos(α-β)= = = ,

所以 sin(α-β)= ,

即 tan (α-β)= , 所以 tan β=tan [α-(α-β)]= 又β∈ 答案: 8.(2012 烟台模拟)已知角α、β的顶点在坐标原点,始边与 x 轴的正 半轴重合,α、β∈(0,π),角β的终边与单位圆交点的横坐标是- , 角α+β的终边与单位圆交点的纵坐标是 ,则 cos α= 解析:依题设得, cos β=- ,∵0<β<π, ∴ <β<π,sin β= , 又∵sin(α+β)= >0,0<α<π, ∴ <α+β<π, cos(α+β)=- . ∴cos α=cos[(α+β)-β]= cos(α+β)cos β+sin(α+β)sin β= -× +× = . ,所以β= . =1,

. 答案: 9.已知 sin 解析:cos 又 cos 所以 cos 答案:三、解答题 10.(2013 宜宾市高三一诊考试)已知函数 f(x)=sin x· cos(π+x). (1)求函数 f(x)的单调区间; (2)在△ABC 中,若 A 为锐角,且 f(A)=1,BC=2,B= ,求 AC 边的长. 解:(1)f(x)=sin =cos2x+sin xcos x =cos2x+ sin 2x = (sin 2x+cos 2x+1) cos x-sin x· cos(π+x) cos x-sin =sin =- . = ,则 cos =2cos2 =, -1, = .

= sin

+.

令- +2kπ<2x+ < +2kπ,k∈Z, 得- +kπ≤x≤kπ+ ,k∈Z, 所以函数 f(x)的单调增区间为 (- +kπ, +kπ),k∈Z, 同理可得函数 f(x)的单调减区间为 ( +kπ, +kπ),k∈Z. (2)因为 f(A)=1, 所以 sin 所以 sin 因为 A 为锐角, 所以 <2A+ < , 所以 2A+ = , 所以 A= . 在△ABC 中,由正弦定理得, = ,即 = , + =1, = .

解得 AC= .

11.(2012 洛阳模拟)已知 cos α= ,cos(α-β)= ,且 0<β<α< . (1)求 tan 2α的值; (2)求β. 解:(1)由 cos α= ,0<α< ,得 sin α= ∴tan α= = = .

= × =4 , =

于是 tan 2α= =- .

(2)由 0<β<α< ,得 0<α-β< , ∵cos(α-β)= , ∴sin(α-β)= 由β=α-(α-β),得 cos β=cos[α-(α-β)]= cos αcos(α-β)+sin αsin(α-β)= × + × =, 所以β= . = = .


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