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2018秋新版高中数学北师大版必修4习题:第一章三角函数 1.4.3 含解析

4.3 单位圆与正弦函数、余弦函数的基本性质 课时过关· 能力提升 π 2π 1.已知函数 y=sin x,x∈ 6 , 3 , 则的取值范围是( A.[-1,1] B. 2 ,1 C. 2 , 2 D. 2 ,1 π π 1 1 3 3 ) 解析:由单位圆可知正弦函数 y=sinx在 6 , 2 上是增加的,在 2 , 3 上是减少的,所以当 x= π 时取得最大值1,当 2 π 2π x= 时取得最小值 . π 6 1 2 答案:B 2.已知函数 y=sin x 的定义域为[a,b],值域为 -1, A. 4π 8π B. C. 2πD. 4π 3 3 2π , ( ? )max 3 1 2 , 则 ? 的最大值和最小值之和等于( ) 解析:利用正弦函数的性质知(b-a)min= 答案:C 3.函数 y= 1 = 4π , 故b-a 的最大值和最小值之和等于 3 2π. 1- 2 sin 的值域是( 1 1 ) A. 2 ,1 B. 0, 2 C. 1, 2 D. 2 , 2 解析:∵1≤sinx≤1,∴ 2≤1? 2 sinx≤2 , 2 ≤ 答案:D 4.函数 y= 1 + 2cos的定义域是( A. - 3 , 3 2π 2π 1 1 3 2 3 2 6 1- 2 sin ≤ 2 . 因此函数y= 1 6 1- 2 sin 的值域是 2 , 2 , 故选D. 1 2 6 ) B. 2π- 3 ,2π + 3 , ∈Z C. - 3 , 3 π π 2π 2π D. 2π- 3 ,2π + 3 , ∈Z 解析:∵1+2cosx≥0,∴cosx≥? 2, ∴x∈ 2π- 3 ,2π + 3 , ∈Z,此即为所求函数的定义域,故选 B. 答案:B 5.函数 y= 1-cos的单调增区间是 . 2π 2π 1 π π 解析:∵y=cosx 的递减区间是[2kπ,2kπ+π],k∈Z,它与 y=-cosx 的单调性相反,∴原函数的递增区间为 [2kπ,2kπ+π],k∈Z. 答案:[2kπ,2kπ+π],k∈Z 6.函数 y= 1+cos 的定义域是 解析:∵1+cosx≠0,∴cosx≠-1, ∴x∈{x|x≠2kπ+π,k∈Z}. ∵2sinx-1≥0,∴sinx≥2, ∴x∈ 2π + 6 ≤ ≤ 2π + 6 ,∈Z . 答案:{x|x≠2kπ+π,k∈Z} 2π + 7.函数 y=-sin x,x∈ - 4 , 3 的最大值为 π 3π π 3π π π 5π 1 1- ; 函数 = 2sin-1的定义域是 . π 5π ≤ ≤ 2π + ,∈Z 6 6 , 最小值为 π . 解析:当 x=? 2 时,函数 y=-sinx,x∈ - 4 , 3 取得最大值1;当 x= 3 时,函数 y=-sinx,x∈ - 4 , 3 取得最小值 ? 2 . 答案:1 ? 3 2 3π π 3 8.函数 y=2-sin x 的值域是 是 答案:[1,3] . ,递增区间是 ,最小正周期 2π + 2 ,2π + 2 (∈Z) 2π .(只填序号) π 3π 9.下列说法正确的有 ①y=|sin x|的定义域为 R; ②y=3sin x+1 的最小值为 1; ③y=sin x-1 的递增区间为 2π + 2 ,2π + 2 (∈Z). 解析:∵y=sinx 的定义域为 R,∴y=|sinx|的定义域为 R,故①正确; 当 sinx=-1 时,ymin=-2,故②错;y=sinx-1 的递增区间为 2π- 2 ,2π + 2 , ∈Z,故③错. 答案:① 10.已知函数 y=asin x+b 的最大值为 0,最小值为-4,求 a,b 的值. 解由题意知, 解得 || + = 0, -|| + = -4, π π π 3π = 2, = -2, 或 = -2 = -2. 3 1 11.若 0<α<2π,求使 sin α< 2 和 cos > 2 同时成立的的范围. 解利用单位圆及正弦函数的性质,在(0,2π)内, 由 sinα< 3 , 得α∈ 2 1 0, π 3 ∪ π 2π ,2 π 3 5π . 同理,由 cosα> 2 , 得α∈ 0, 3 ∪ 3 ,2π . 故所求 α 的范围是 0, 3 ∪ 3 ,2π . ★12.函数 f(x)=-sin2x+sin x+a,若 1≤f(x)≤ 4 对一切∈R 恒成立,求 a 的取值范围. 解 f(x)=? sin- 2 1 1 2 17 π 5π + 4 + , 当sinx=-1 时,ymin=a-2; 1 1 当 sinx= 2 时,ymax= 4 + , ∴f(x)的值域为 -2, + . -2 ≥ 1, ≥ 3, ∴ 1 17 ∴ ≤ 4, 即3≤a≤4. + ≤ 4 , 4 ∴a 的取值范围是[3,4]. 1 4

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