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www.77640.com:高中数学北师大版必修4第1章5《正弦函数的图像与性质》word课时作业

【成才之路】2015-2016 学年高中数学 第 1 章 5 正弦函数的图像与 性质课时作业 北师大版必修 4 一、选择题 π 1.下列两种说法:①y=sinx 在[2kπ - ,2kπ ](k∈Z)上是增加的;②y=sinx 在第 2 一象限内是增加的( A.均正确 C.②正确、①不正确 [答案] B [解析] 单调性是针对某个取值区间而言的, 所以①正确; ②不正确, 因为在第一象限, 即使是终边相同的角,它们也相差 2π 的整数倍. 2.下列函数具有奇偶性的是( A.y=sinx(x>0) 1 C.y=sin (x≠0) ) B.y=2sinx(x<0) D.y= 2sinx ) B.①正确、②不正确 D.都不正确 x [答案] C [解析] 对于选项 A,定义域为(0,+∞),函数图像不关于原点对称. 对于选项 B,定义域为(-∞,0),函数图像也不关于原点对称. 1 对于选项 C,定义域为(-∞,0)∪(0,+∞)关于原点对称,并且 f(-x)=sin(- ) x 1 =-sin =-f(x),所以为奇函数. x 对于选项 D,定义域不关于原点对称. 3 3.y=1+sinx,x∈[0,2π ]的图像与 y= 交点的个数是( 2 A.0 C.2 [答案] C 3 [解析] 如图,y=1+sinx,x∈[0,2π ]的图像与 y= 的图像有两个交点. 2 B.1 D.3 ) 4.函数 y=|sinx|的一个单调增区间是( π π A.(- , ) 4 4 3π C.(π , ) 2 [答案] C ) π 3π B.( , ) 4 4 3π D.( ,2π ) 2 [解析] 画出 y=|sinx|的图像即可解决.借助图像不难看出 C 符合题意. 1 5.在[0,2π ]上满足 sinx≥ 的 x 的取值范围是( 2 ) ? π? A.?0, ? 6? ? ?π 2 ? C.? , π ? ?6 3 ? [答案] B ?π 5 ? B.? , π ? ?6 6 ? ?5 ? D.? π ,π ? ?6 ? ?π 5 ? [解析] 如图可知 x∈? , π ? ?6 6 ? π 6.点 M( ,b)在函数 y= 2sinx+1 的图像上,则 b 等于( 4 A. 2 2 B. 2 D.3 ) C.2 [答案] C π π [解析] b=f( )= 2sin +1=2. 4 4 二、填空题 7.函数 y=sin x-2sinx 的值域是________. [答案] [-1,3] [解析] y=(sinx-1) -1,∵-1≤sinx≤1, ∴-2≤sinx-1<1, 2 2 ∴0≤(sinx-1) ≤4,可得-1≤y≤3. 8.函数 y=lg sin 的定义域是________. 2 [答案] [4kπ ,4kπ +2π ],k∈Z [解析] 由 sin >0,得 2kπ < <2kπ +π ,k∈Z,解得 4kπ <x<4kπ +2π ,k∈Z. 2 2 三、解答题 1 ? π 5π ? 2 9.求函数 f(x)=2sin x+2sinx- ,x∈? , ?的值域. 6 ? 2 ?6 2 x x x ? π 5π ? [解析] 令 t=sinx,因为 x∈? , ?, 6 ? ?6 1 1 所以 ≤sinx≤1,即 ≤t≤1. 2 2 1 1 2 1 1 2 ∴y=2t +2t- =2(t+ ) -1,t∈[ ,1],且该函数在[ ,1]上单调递增. 2 2 2 2 1 7 ∴f(x)最小值为 f( )=1,最大值为 f(1)= . 2 2 ? 7? ∴f(x)的值域为?1, ?. ? 2? 10.比较大小: π 2π (1)sin 与 sin ; 4 3 (2)sin(-320°)与 sin700°. 2π π π [解析] (1)∵sin =sin(π - )=sin , 3 3 3 0< π π π π < < ,y=sinx 在(0, )上是增加的, 4 3 2 2 π π π 2π ∴sin <sin ,即 sin <sin . 4 3 4 3 (2)∵sin(-320°)=sin(-360°+40°)=sin40°, sin700°=sin(720°-20°)=sin(-20°). π π 又函数 y=sinx 在[- , ]上是增加的, 2 2 ∴sin40°>sin(-20°),即 sin(-320°)>sin700°. 一、选择题 1. 设点 P 是函数 f(x)=sinω x 的图像 C 的一个对称中心, 若点 P 到图像 C 的对称轴的 π 距离的最小值为 ,则 f(x)的最小正周期为( 4 A.2π π C. 2 [答案] B ) B.π π D. 4 [解析] 函数 f(x)=sinω x 与 f(x)=sinx 的图像形状相同, 观察图像可知对称中心与 1 1 π 对称轴最近距离为 T.由题意得 T= ,所以 T=π . 4 4 4 2.方程 sinx=lgx 的解的个数是( A.0 C.2 [答案] D [解析] 利用图像可知 y=sinx 与 y=lgx 的图像有 3 个交点. ) B.1 D.3 二、填空题 3.函数 y=lg(3-4sin x)的定义域是______________. π π? ? [答案] ?kπ - ,kπ + ?,k∈Z 3 3? ? [解析] 3-4sin x>0,解得- 2 2 3 3 <sinx< , 2 2 π π? ? ∴x∈?kπ - ,kπ + ?,k∈Z. 3 3? ? 4.下列说法正确的有________(只填序号). ①y=|sinx|的定义域为 R; ②y=3sinx+1 的最小值为 1; ③y=-sinx 为奇函数; π 3π ④y=sinx-1 的单调递增区间为[2kπ + ,2kπ + ](k∈R). 2 2 [答案] ①③ [解析] 对于②,y=

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