录入空中课堂三角函数例题与习题
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#修改起始id,出处,文件名
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starting_id = 18315 #起始id设置, 来自"寻找空闲题号"功能
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starting_id = 18434 #起始id设置, 来自"寻找空闲题号"功能
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raworigin = "" #题目来源的前缀(中缀在.tex文件中)
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filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\空中课堂必修第二册例题与习题.tex" #题目的来源.tex文件
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editor = "王伟叶" #编辑者姓名
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@ -474987,6 +474987,726 @@
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"space": "4em",
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"unrelated": []
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},
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"018434": {
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"id": "018434",
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"content": "用``五点法''作出函数$y=1-\\sin x$, $x \\in[0,2 \\pi]$的大致图像, 并写出使得$y<1$的$x$的取值范围.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018435": {
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"id": "018435",
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"content": "作出函数$y=2-\\sin x$, $x \\in[-\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2}]$的大致图像.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018436": {
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"id": "018436",
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"content": "设$a$为常数, 若满足$a=\\sin x-\\dfrac{1}{2}$且$x \\in[-\\pi, \\pi]$的$x$的值只有一个, 则实数$a$的值为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018437": {
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"id": "018437",
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"content": "满足$2 \\sin |x|-\\sqrt{3}>0$且$x \\in[-2 \\pi, 2 \\pi]$的$x$的取值集合为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018438": {
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"id": "018438",
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"content": "若函数$y=\\sin x+2|\\sin x|, x \\in[0,2 \\pi]$的图像与直线$y=k$有且只有两个公共点, 则实数$k$的取值范围为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018439": {
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"id": "018439",
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"content": "求下列函数$y=f(x)$的最小正周期:\\\\\n(1) $f(x)=\\sin 3 x$;\\\\\n(2) $f(x)=2 \\sin (-\\dfrac{1}{2} x+\\dfrac{\\pi}{3})$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018440": {
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"id": "018440",
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"content": "已知函数$y=\\sin (k x+\\dfrac{\\pi}{3}$) (其中常数$k \\neq 0$) 的最小正周期是 $2$ , 求$k$的值.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018441": {
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"id": "018441",
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"content": "函数$y=|\\sin x|$的最小正周期为\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018442": {
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"id": "018442",
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"content": "已知$f(n)=\\sin \\dfrac{\\pi n}{6}$, $n \\in \\mathbf{N}$, 求$f(1)+f(2)+f(3)+\\cdots+f(2021)$的值.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018443": {
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"id": "018443",
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"content": "求下列函数的最大值和最小值, 并求出取得最大值和最小值时所有$x$的值:\\\\\n(1) $y=-2 \\sin (3 x+\\dfrac{\\pi}{3})$;\\\\\n(2) $y=\\sin x+\\sqrt{3} \\cos x$;\\\\\n(3) $y=\\sin ^2 x-\\sin x$;\\\\\n(4) $y=\\sin ^2 x+2 \\sqrt{3} \\sin x \\cos x+3 \\cos ^2 x$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018444": {
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"id": "018444",
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"content": "如图所示, 在一个半径为$r$的半圆形铁板中, 截取一块矩形$ABCD$, 使得矩形的顶点$A$、$B$在半圆的直径上, $C$、$D$在半圆弧上. 问: 如何截取矩形$ABCD$, 使其面积达到最大值? 并求出这个最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) coordinate (S) (2,0) coordinate (T);\n\\draw (S) arc (180:0:2);\n\\filldraw (0,0) circle (0.03) node [below] {$O$} coordinate (O);\n\\draw (40:2) node [above right] {$C$} coordinate (C);\n\\draw (140:2) node [above left] {$D$} coordinate (D);\n\\draw ($(S)!(D)!(T)$) node [below] {$A$} coordinate (A);\n\\draw ($(S)!(C)!(T)$) node [below] {$B$} coordinate (B);\n\\draw (S)--(T)(A)--(D)--(C)--(B);\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018445": {
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"id": "018445",
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"content": "求函数$y=|\\sin x|+|\\cos x|$的最大值和最小值, 并求使其取得最大值和最小值的$x$的集合.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018446": {
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"id": "018446",
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"content": "如图, 某小区有一块半径为$10$米的扇形空地, 其圆心角$\\angle AOB=\\dfrac{\\pi}{3}$, 现规划在此空地上建造一块矩形的健身区域$CDEF$, 其中点$C$、$D$在半径$OA$上, 点$E$在$\\overset\\frown{AB}$上, 点$F$在半径$OB$上, 其余地方进行绿化. 设$\\angle EOA=\\theta$, 当$\\theta$为何值时, 健身区域$CDEF$的面积$S$最大? 并求出面积$S$的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (40:2) node [above right] {$E$} coordinate (E);\n\\draw ($(O)!(E)!(A)$) node [below] {$D$} coordinate (D);\n\\path [name path = OB] (O) -- (B);\n\\path [name path = EF] (E) --++ (-1.5,0);\n\\path [name intersections = {of = OB and EF, by = F}];\n\\draw (F) node [above left] {$F$};\n\\draw ($(O)!(F)!(A)$) node [below] {$C$} coordinate (C);\n\\draw (C)--(F)--(E)--(D)(A)--(O)--(B)(A) arc (0:60:2)(O)--(E);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 1.8] {angle = A--O--E};\n\\end{tikzpicture}\n\\end{center}",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018447": {
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"id": "018447",
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"content": "判断下列函数$y=f(x)$的奇偶性, 并说明理由:\\\\\n(1) $f(x)=\\sin |x|$;\\\\\n(2) $f(x)=\\sin (x+\\dfrac{\\pi}{2})$;\\\\\n(3) $f(x)=\\sin (x-\\dfrac{\\pi}{4})$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018448": {
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"id": "018448",
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"content": "利用函数的单调性, 比较下列各组数的大小:\\\\\n(1) $\\sin \\dfrac{6 \\pi}{5}$与$\\sin \\dfrac{7 \\pi}{6}$;\\\\\n(2) $\\sin \\dfrac{43 \\pi}{7}$与$\\sin (-\\dfrac{47 \\pi}{8})$.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018449": {
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"id": "018449",
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"content": "求函数$y=\\sin (x+\\dfrac{\\pi}{2})$的单调减区间.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018450": {
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"id": "018450",
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"content": "求函数$y=2 \\sin (-2 x+\\dfrac{\\pi}{6})$, $x \\in(-\\pi, 0]$的单调增区间.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "4em",
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"unrelated": []
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},
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"018451": {
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"id": "018451",
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"content": "设$\\varphi$为给定的实数. 函数$y=4 \\sin (2 x+\\varphi)$为偶函数的一个充要条件是$\\varphi=$\\blank{50}.",
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"objs": [],
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"tags": [],
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"genre": "填空题",
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"ans": "",
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"solution": "",
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"duration": -1,
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"usages": [],
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"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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||||
"20230705\t王伟叶"
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],
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"same": [],
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"related": [],
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"remark": "",
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"space": "",
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"unrelated": []
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},
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"018452": {
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"id": "018452",
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"content": "已知函数$y=2 \\sqrt{3} \\sin x \\cos x+\\cos ^2 x-\\sin ^2 x-1, x \\in[0, \\dfrac{7 \\pi}{12}]$.\\\\\n(1) 求此函数的单调增区间;\\\\\n(2) 求此函数的值域.",
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"objs": [],
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"tags": [],
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"genre": "解答题",
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"ans": "",
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"solution": "",
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"duration": -1,
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||||
"usages": [],
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||||
"origin": "空中课堂必修第二册三角函数例题与习题",
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"edit": [
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||||
"20230705\t王伟叶"
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||||
],
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"same": [],
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||||
"related": [],
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"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018453": {
|
||||
"id": "018453",
|
||||
"content": "求下列函数的最大值与最小值, 并求出取得最大值和最小值时所有$x$的值:\\\\\n(1) $y=\\cos ^2 x-4 \\cos x+1$, $x \\in \\mathbf{R}$;\\\\\n(2) $y=\\cos \\dfrac{x}{2}$, $x \\in[-\\dfrac{4 \\pi}{3}, \\dfrac{\\pi}{2}]$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018454": {
|
||||
"id": "018454",
|
||||
"content": "求函数$y=2 \\cos (2 x-\\dfrac{\\pi}{3})$的最小正周期及单调增区间.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018455": {
|
||||
"id": "018455",
|
||||
"content": "设常数$\\omega>0$, 求函数$y=\\cos ^2 \\omega x$的最小正周期和单调区间.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018456": {
|
||||
"id": "018456",
|
||||
"content": "观察余弦曲线, 分别写出满足下列条件的所有$x$的集合:\\\\\n(1) $\\cos x<0$;\\\\\n(2) $\\cos x>-\\dfrac{\\sqrt{2}}{2}$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018457": {
|
||||
"id": "018457",
|
||||
"content": "当函数$y=A \\sin (\\omega x+\\varphi)$($A>0$, $\\omega>0$)中的常数$A$、$\\omega$、$\\varphi$分别取下列各组值时, 用计算器 (机) 在同一平面直角坐标系中作出它们的图像:\\\\\n(1) $A=2, \\omega=1, \\varphi=0$;\\\\\n(2) $A=1, \\omega=2, \\varphi=0$;\\\\\n(3) $A=1, \\omega=1, \\varphi=\\dfrac{\\pi}{2}$.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018458": {
|
||||
"id": "018458",
|
||||
"content": "作出函数$y=3 \\sin (2 x+\\dfrac{\\pi}{4})$的大致图像, 并指出其振幅、频率和初始相位.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018459": {
|
||||
"id": "018459",
|
||||
"content": "已知交流电的电流强度$I$关于时间$t$的函数为$I=I_0 \\sin (\\omega t+\\varphi)$, 其中$I_0>0$, $\\omega>0$, $0 \\leq \\varphi<2 \\pi$. 根据图像求出它的周期、频率和电流的最大值, 并写出$I_0$、$\\omega$和$\\varphi$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 1.5, yscale = 0.08]\n\\draw [->] (0,0) -- (4,0) node [below] {$t$(s)};\n\\draw [->] (0,-15) -- (0,15) node [left] {$I$(A)};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0:3, samples = 100] plot (\\x,{10*sin(\\x*180)});\n\\foreach \\i/\\j in {0.01/below left,0.02/below right,0.03/below}\n{\\draw ({\\i*100},2) --++ (0,-2) node [\\j] {\\small$\\i$};};\n\\draw [dashed] (3.5,10) -- (0,10) node [left] {\\small$10$} (3.5,-10) -- (0,-10) node [left] {\\small$-10$};\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018460": {
|
||||
"id": "018460",
|
||||
"content": "函数$y=2 \\sin 3 x$, $x \\in[\\dfrac{\\pi}{6}, \\dfrac{5 \\pi}{6}]$与函数$y=2$的图像所围成的封闭图形的面积为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018461": {
|
||||
"id": "018461",
|
||||
"content": "已知函数$y=\\sin (\\omega x+\\varphi)$(其中常数$\\omega$、$\\varphi$满足$\\omega>0$且$0 \\leq \\varphi \\leq \\pi$) 是$\\mathbf{R}$上的偶函数, 其图像关于点$P(\\dfrac{2 \\pi}{3}, 0)$成中心对称, 且在区间$[0, \\dfrac{\\pi}{2}]$上是单调函数, 求$\\varphi$和$\\omega$的值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018462": {
|
||||
"id": "018462",
|
||||
"content": "求函数$y=\\tan (\\dfrac{\\pi}{6} x+\\dfrac{\\pi}{3})$的定义域和单调区间.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018463": {
|
||||
"id": "018463",
|
||||
"content": "求函数$y=\\tan (2 x+\\dfrac{\\pi}{6})$的最小正周期.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018464": {
|
||||
"id": "018464",
|
||||
"content": "函数$y=2 \\tan x+\\tan (\\dfrac{\\pi}{2}-x)$, $x \\in(0, \\dfrac{\\pi}{2})$的最小值为\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018465": {
|
||||
"id": "018465",
|
||||
"content": "已知函数$y=2 \\tan (n x-\\dfrac{\\pi}{3})$的最小正周期$T$满足$1<T<\\dfrac{3}{2}$, 其中$n$为正整数, 求$n$的值, 并写出该函数的单调区间.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018466": {
|
||||
"id": "018466",
|
||||
"content": "已知函数$y=\\sqrt{3} \\sin \\dfrac{x}{2}+\\cos \\dfrac{x}{2}$($x \\in \\mathbf{R}$). 用``五点法''画出它的大致图像.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018467": {
|
||||
"id": "018467",
|
||||
"content": "设常数$A>0$, $\\omega>0$, $0<\\varphi<\\pi$, 函数$y=A \\sin (\\omega x+\\varphi)$的部分图像如图所示.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-2.5) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [above right] {$O$};\n\\draw [domain = -1:{2*pi/3}, samples = 200] plot (\\x,{2*sin(2*\\x/pi*180+120)});\n\\filldraw ({pi/6},0) circle (0.03) node [below left] {\\small$\\dfrac{\\pi}{6}$};\n\\filldraw ({-pi/12},0) circle (0.03) node [below] {\\small$\\dfrac{\\pi}{12}$};\n\\filldraw (0,2) circle (0.03) node [right] {\\small$2$};\n\\draw [dashed] ({-pi/12},0) --++ (0,2) --(0,2);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A, \\omega, \\varphi$的值;\\\\\n(2) 求函数$y=A \\sin (\\omega x+\\varphi)$, $x \\in[0, \\dfrac{\\pi}{2}]$的最值.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"018468": {
|
||||
"id": "018468",
|
||||
"content": "设$\\omega$为正数, 若函数$y=\\sin \\omega x$在区间$[-\\dfrac{\\pi}{4}, \\dfrac{2 \\pi}{3}]$上为严格增函数, 则$\\omega$的取值范围是\\blank{50}.",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "填空题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "",
|
||||
"unrelated": []
|
||||
},
|
||||
"018469": {
|
||||
"id": "018469",
|
||||
"content": "如图, 某公园有一块半径为$r$米的扇形绿地$AOB$, 其中$O$为扇形所在圆的圆心, $\\angle AOB=60^{\\circ}$. 欲在该绿地上修建小路$CD$、$CE$, 点$C$在$\\overset\\frown{AB}$上, $D$、$E$分别在半径$OA$、$OB$上, 且$CD\\parallel OB$, $CE\\parallel OA$, 小路$CD$与$CE$的总长为$l$米, 设$\\angle AOC=\\theta$. 试将$l$表示为$\\theta$的函数, 并求$l$的最大值及取得最大值时$\\theta$的值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (2,0) node [below] {$A$} coordinate (A);\n\\draw (60:2) node [above] {$B$} coordinate (B);\n\\draw (40:2) node [above right] {$C$} coordinate (C);\n\\path [name path = OB] (O) -- (B);\n\\path [name path = CE] (C) --++ (-1.5,0);\n\\path [name intersections = {of = OB and CE, by = E}];\n\\draw (E) node [above left] {$E$};\n\\path [name path = OA] (O) -- (A);\n\\path [name path = CD] (C) --++ (-120:1.5);\n\\path [name intersections = {of = OA and CD, by = D}];\n\\draw (D) node [below] {$D$};\n\\draw (D)--(C)--(E)(O)--(C)(A)--(O)--(B)(A)arc(0:60:2);\n\\draw pic [draw, \"$\\theta$\", scale = 0.6, angle eccentricity = 1.8] {angle = A--O--C};\n\\end{tikzpicture}\n\\end{center}",
|
||||
"objs": [],
|
||||
"tags": [],
|
||||
"genre": "解答题",
|
||||
"ans": "",
|
||||
"solution": "",
|
||||
"duration": -1,
|
||||
"usages": [],
|
||||
"origin": "空中课堂必修第二册三角函数例题与习题",
|
||||
"edit": [
|
||||
"20230705\t王伟叶"
|
||||
],
|
||||
"same": [],
|
||||
"related": [],
|
||||
"remark": "",
|
||||
"space": "4em",
|
||||
"unrelated": []
|
||||
},
|
||||
"020001": {
|
||||
"id": "020001",
|
||||
"content": "判断下列各组对象能否组成集合, 若能组成集合, 指出是有限集还是无限集.\\\\\n(1) 上海市控江中学$2022$年入学的全体高一年级新生;\\\\\n(2) 中国现有各省的名称;\\\\\n(3) 太阳、$2$、上海市;\\\\\n(4) 大于$10$且小于$15$的有理数;\\\\\n(5) 末位是$3$的自然数;\\\\\n(6) 影响力比较大的中国数学家;\\\\\n(7) 方程$x^2+x-3=0$的所有实数解;\\\\ \n(8) 函数$y=\\dfrac 1x$图像上所有的点;\\\\ \n(9) 在平面直角坐标系中, 到定点$(0, 0)$的距离等于$1$的所有点;\\\\\n(10) 不等式$3x-10<0$的所有正整数解;\\\\\n(11) 所有的平面四边形.",
|
||||
|
|
|
|||
Reference in New Issue