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2023-2024学年人教A版数学必修第一册达标自测5.4.1正弦函数、余弦函数的图象（含解析）

2023-11-11 12:21:16 学考宝作者：佚名

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第五章　5.4.1正弦函数、余弦函数的图象
一、选择题
1．已知正弦函数过点，则m的值为(　　)
A． B．－　
C． D．1
2．用“五点法”画函数y＝1＋sin x的图象时，首先应描出五点的横坐标是(　　)
A．0，，，，π B．0，，π，，2π
C．0，π，2π，3π，4π D．0，，，，
3．函数y＝1－sin x，x∈[0，2π]的大致图象是(　　)
INCLUDEPICTURE "H:\\111111\\111111\\W11A．tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W11A.tif" \* MERGEFORMATINET
4．在[0，2π]上满足sin x≥的x的取值范围是(　　)
A． B．
C． D．
5．如图所示的曲线对应的函数解析式可以是下列选项中的(　　)
INCLUDEPICTURE "H:\\111111\\111111\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W17.tif" \* MERGEFORMATINET
A．y＝|sin x| B．y＝sin |x|
C．y＝－sin |x| D．y＝－|sin x|
6．函数y＝cos x＋|cos x|，x∈[0，2π]的大致图象为(　　)
INCLUDEPICTURE "H:\\111111\\111111\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\20Y52.TIF" \* MERGEFORMATINET
7．若cos x＝0，则角x等于(　　)
A．kπ(k∈Z)
B．＋kπ(k∈Z)
C．＋2kπ(k∈Z)
D．－＋2kπ(k∈Z)
8．当x∈[0，2π]时，满足sin≥－的x的取值范围是(　　)
A． B．
C．∪ D．
9．(多选题)下列在(0，2π)上的区间能使cos x>sin x成立的是(　　)
A．
B．
C．
D．∪
10．(多选题)若函数f(x)＝2cos x(0≤x≤2π)的图象和直线y＝2围成一个封闭的平面图形，则下列说法正确的是(　　)
A．当x∈时，y<0
B．f(0)＝1
C．f＝0
D．阴影部分的面积为2π
二、填空题
11．已知函数f(x)＝3＋2cos x的图象经过点，则b＝．
12．方程x2＝cos x的实根个数是．
INCLUDEPICTURE "H:\\111111\\111111\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W15.tif" \* MERGEFORMATINET
13．若方程sin x＝4m＋1在x∈[0，2π]上有解，则实数m的取值范围是．
14．函数y＝lg(1－2sin x)的定义域是．
15．函数y＝sin x－1，x∈[0，2π]与y＝a有一个交点，则a的值为
三、解答题
16．利用“五点法”作出下列函数的简图．
(1)y＝2sin x－1(0≤x≤2π)；
(2)y＝－1－cos x(0≤x≤2π)．
17．已知函数f(x)＝
(1)作出该函数的图象；
(2)若f(x)＝，求x的值．
18．观察y＝sin x，x∈R的图象，回答下列问题：
(1)当x从0变到时，sin x的值增大还是减小？是正的还是负的？
(2)对应于x＝，sin x有多少个值？
(3)对应于sin x＝，x有多少个值？并写出x的值．
19．若方程sin x＝在x∈上有两个实数根，求a的取值范围．
第五章　5.4.1正弦函数、余弦函数的图象
一、选择题
1．已知正弦函数过点，则m的值为(　A　)
A． B．－　
C． D．1
2．用“五点法”画函数y＝1＋sin x的图象时，首先应描出五点的横坐标是(　B　)
A．0，，，，π B．0，，π，，2π
C．0，π，2π，3π，4π D．0，，，，
[解析]　利用五点法作图的要求可知，选B．
3．函数y＝1－sin x，x∈[0，2π]的大致图象是(　B　)
INCLUDEPICTURE "H:\\111111\\111111\\W11A．tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W11A.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W11A.tif" \* MERGEFORMATINET
[解析]　利用代入特殊值法即可得出选B．
4．在[0，2π]上满足sin x≥的x的取值范围是(　B　)
A． B．
C． D．
[解析]　由函数y＝sin x，x∈[0，2π]的图象，可知≤x≤.
5．如图所示的曲线对应的函数解析式可以是下列选项中的(　C　)
INCLUDEPICTURE "H:\\111111\\111111\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W17.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W17.tif" \* MERGEFORMATINET
A．y＝|sin x| B．y＝sin |x|
C．y＝－sin |x| D．y＝－|sin x|
[解析]　将代入4个解析式，排除A，B；将代入C，D中的解析式，排除D，故选C．
6．函数y＝cos x＋|cos x|，x∈[0，2π]的大致图象为(　D　)
INCLUDEPICTURE "H:\\111111\\111111\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\20Y52.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\20Y52.TIF" \* MERGEFORMATINET
[解析]　y＝cos x＋|cos x|
＝故选D．
7．若cos x＝0，则角x等于(　B　)
A．kπ(k∈Z)
B．＋kπ(k∈Z)
C．＋2kπ(k∈Z)
D．－＋2kπ(k∈Z)
8．当x∈[0，2π]时，满足sin≥－的x的取值范围是(　C　)
A． B．
C．∪ D．
[解析]　由诱导公式化简可得cos x≥－，结合余弦函数的图象可知选C．
9．(多选题)下列在(0，2π)上的区间能使cos x>sin x成立的是(　AC　)
A．
B．
C．
D．∪
[解析]　在同一平面直角坐标系中画出正、余弦函数的图象，在(0，2π)上，当cos x＝sin x时，x＝或x＝，结合图象可知满足cos x>sin x的是和，故选AC．
INCLUDEPICTURE "H:\\111111\\111111\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\AB76.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\AB76.TIF" \* MERGEFORMATINET
10．(多选题)若函数f(x)＝2cos x(0≤x≤2π)的图象和直线y＝2围成一个封闭的平面图形，则下列说法正确的是(　AC　)
A．当x∈时，y<0
B．f(0)＝1
C．f＝0
D．阴影部分的面积为2π
[解析]　作出函数y＝2cos x，x∈[0，2π]的图象，函数y＝2cos x，x∈[0，2π]的图象与直线y＝2围成的平面图形为如图所示的阴影部分，由图可知，A正确；B错误；C正确；
INCLUDEPICTURE "H:\\111111\\111111\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\AB77.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\AB77.TIF" \* MERGEFORMATINET
利用图象的对称性，可知该阴影部分的面积等于矩形OABC的面积，又∵OA＝2，OC＝2π，
∴S阴影部分＝S矩形OABC＝2×2π＝4π，∴D错误．故选AC．
二、填空题
11．已知函数f(x)＝3＋2cos x的图象经过点，则b＝4．
[解析]　b＝f＝3＋2cos＝4.
12．方程x2＝cos x的实根个数是2．
[解析]　画出y＝x2和y＝cos x的图象如图所示，观察交点个数为2.
INCLUDEPICTURE "H:\\111111\\111111\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W15.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W15.tif" \* MERGEFORMATINET
13．若方程sin x＝4m＋1在x∈[0，2π]上有解，则实数m的取值范围是．
[解析]　由正弦函数的图象，知当x∈[0，2π]时，sin x∈[－1，1]，要使得方程sin x＝4m＋1在x∈[0，2π]上有解，则－1≤4m＋1≤1，故－≤m≤0.
14．函数y＝lg(1－2sin x)的定义域是．
[解析]　由题意可得，函数y＝lg(1－2sin x)满足1－2sin x>0，即sin x<.由正弦函数的图象知，sin x<在上的解集为，所以在R上的解集为，k∈Z，故函数y＝lg(1－2sin x)的定义域为.
15．函数y＝sin x－1，x∈[0，2π]与y＝a有一个交点，则a的值为0或－2．
[解析]　y＝sin x－1的图象是由y＝sin x的图象向下平移1个单位得到的，其最高点为，最低点为，所以a的值为0或－2.
三、解答题
16．利用“五点法”作出下列函数的简图．
(1)y＝2sin x－1(0≤x≤2π)；
(2)y＝－1－cos x(0≤x≤2π)．
[解析]　(1)列表：
x 0 π 2π
2sin x 0 2 0 －2 0
2sin x－1 －1 1 －1 －3 －1
描点作图，如图所示：
INCLUDEPICTURE "H:\\111111\\111111\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W18.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W18.tif" \* MERGEFORMATINET
(2)列表：
x 0 π 2π
cos x 1 0 －1 0 1
－1－cos x －2 －1 0 －1 －2
描点作图，如图所示．
INCLUDEPICTURE "H:\\111111\\111111\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W19.tif" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\W19.tif" \* MERGEFORMATINET
17．已知函数f(x)＝
(1)作出该函数的图象；
(2)若f(x)＝，求x的值．
[解析]　(1)作出函数f(x)＝的图象，如图①所示．
INCLUDEPICTURE "H:\\111111\\111111\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "H:\\111111\\111111\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "D:\\MMMMMMMMM\\原稿\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\1124SXM16.TIF" \* MERGEFORMATINET INCLUDEPICTURE "C:\\Users\\Administrator\\Desktop\\成才之路数学必修第一册RJA\\新建文件夹\\1124SXM16.TIF" \* MERGEFORMATINET
(2)因为f(x)＝，所以在图①基础上再作直线y＝，如图②所示，则当－π≤x<0时，由图象知x＝－，当0≤x≤π时，x＝或x＝.
综上，可知x的值为－或或.
18．观察y＝sin x，x∈R的图象，回答下列问题：
(1)当x从0变到时，sin x的值增大还是减小？是正的还是负的？
(2)对应于x＝，sin x有多少个值？
(3)对应于sin x＝，x有多少个值？并写出x的值．
[解析]　根据图象可得，
(1)当x从0变到时，sin x的值增大，且是正的．
(2)对应于x＝，sin x有一个值，为.
(3)对应于sin x＝，x有无数个值，且x＝2kπ＋或x＝2kπ＋(k∈Z)．
19．若方程sin x＝在x∈上有两个实数根，求a的取值范围．
[解析]　首先作出y＝sin x，x∈的图象，然后再作出y＝的图象，如果y＝sin x，x∈与y＝的图象有两个交点，方程sin x＝，x∈就有两个实数根．设y1＝sin x，x∈，y2＝.y1＝sin x，x∈的图象如图．
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由图象可知，当≤<1，即－1