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《步步高 学案导学设计》2013-2014学年 高中数学人教B版必修4第一章 1.2.2单位圆与三角函数线

1.2.2
一、基础过关

单位圆与三角函数线

π 5π π 4π π 5π 1. 有三个命题:① 和 的正弦线长度相等;② 和 的正切线相同;③ 和 的余弦线长 6 6 3 3 4 4 度相等. 其中正确说法的个数为 A.1 B.2 C.3 D.0 ( ) ( )

2. 利用正弦线比较 sin 1,sin 1.2,sin 1.5 的大小关系是 A.sin 1>sin 1.2>sin 1.5 B.sin 1>sin 1.5>sin 1.2 C.sin 1.5>sin 1.2>sin 1 D.sin 1.2>sin 1>sin 1.5 π 3. 函数 y=tan?x-3?的定义域为 ? ? π ? ? A.?x|x≠3,x∈R?
? ? ? ? ?

(

)

π ? ? B.?x|x≠kπ+6,k∈Z?
? ?

5π ? ? C.?x|x≠kπ+ 6 ,k∈Z? 5π ? ? D.?x|x≠kπ- 6 ,k∈Z?
?

4. 设 a=sin(-1),b=cos(-1),c=tan(-1),则有 A.a<b<c C.c<a<b 5. 若 0<α<2π,且 sin α< π π A.?-3,3? ? ? 5π C.? 3 ,2π? ? ? π π 6. 如果 <α< ,那么下列不等式成立的是 4 2 A.cos α<sin α<tan α C.sin α<cos α<tan α B.tan α<sin α<cos α D.cos α<tan α<sin α B.b<a<c D.a<c<b 3 1 ,cos α> ,则角 α 的取值范围是 2 2 π B.?0,3? ? ? π 5π D.?0,3?∪? 3 ,2π? ? ? ? ?

(

)

(

)

(

)

7. 集合 A=[0,2π],B={α|sin α<cos α},则 A∩B=________________.

8. 利用三角函数线,写出满足下列条件的角 x 的集合: 1 1 (1)sin x>- 且 cos x> ;(2)tan x≥-1. 2 2

二、能力提升 9. 不等式 tan α+ 3 >0 的解集是______________. 3

10.求函数 f(x)= cos2x-sin2x的定义域为_______________. θ θ θ 11.设 θ 是第二象限角,试比较 sin ,cos ,tan 的大小. 2 2 2 π 12.设 >α>β>0,求证:α-β>sin α-sin β. 2

三、探究与拓展 π 13.当 α∈?0,2?时,求证:sin α<α<tan α. ? ?

答案 1.C 2.C 3.C 4.C 5.D 6.A π 5 7.?0,4?∪?4π,2π? ? ? ? ? 8. 解

π π 1 1 ? ? ? (1)由图(1)知: sin x>- 且 cos x> 时, x 满足的集合:x|-6+2kπ<x<3+2kπ,k∈Z?. 当 角 2 2 ? ? (2)由图(2)知:当 tan x≥-1 时,角 x 满足的集合: π π ? ? ?x|2kπ- ≤x<2kπ+ ,k∈Z?∪ 4 2 ? ? 3 3 ? ? ?x|2kπ+ π≤x<2kπ+ π,k∈Z?, 4 2 ? ? π π ? ? 即?x|nπ-4≤x<nπ+2,n∈Z?. ? ? π π ? ? 9.?α|kπ-6<α<kπ+2,k∈Z?
? ?

π π 10.?kπ-4,kπ+4?,k∈Z ? ? 11.解 θ 是第二象限角,

π 即 2kπ+ <θ<2kπ+π (k∈Z), 2 π θ π 故 kπ+ < <kπ+ (k∈Z). 4 2 2 θ 作出 所在范围如图所示. 2 π θ π 当 2kπ+ < <2kπ+ (k∈Z)时, 4 2 2

θ θ θ 易知 OM<MP<AT.∴cos <sin <tan ; 2 2 2 5 θ 3 当 2kπ+ π< <2kπ+ π(k∈Z)时, 4 2 2

θ θ θ 易知 MP<OM<AT.∴sin <cos <tan . 2 2 2 12.证明 如图所示,设单位圆与角 α、β 的终边分别交于 P1、P2,

作 P1M1⊥x 轴于 M1, P2M2⊥x 轴于 M2, P2C⊥P1M1 于 C, 作 作 连接 P1P2,则 sin α=M1P1,sin β=M2P2,α-β= P1P2 , ∴α-β= P1P2 >P1P2>CP1=M1P1-M1C=M1P1-M2P2=sin α -sin β,即 α-β>sin α-sin β. 13.证明 如图所示,在直角坐标系中作出单位圆,α 的终边与单位圆 交于 P,α 的正弦线、正切线为有向线段 MP,AT,则 MP=sin α, AT=tan α. 1 1 因为 S△AOP= OA· MP= sin α, 2 2 1 1 S 扇形 AOP= αOA2= α, 2 2 1 1 S△AOT= OA· AT= tan α,又 S△AOP<S 扇形 AOP<S△AOT, 2 2 1 1 1 所以 sin α< α< tan α,即 sin α<α<tan α. 2 2 2


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