diff --git a/工具v2/批量收录题目.py b/工具v2/批量收录题目.py
index d61327fa..e0560750 100644
--- a/工具v2/批量收录题目.py
+++ b/工具v2/批量收录题目.py
@@ -1,9 +1,9 @@
 #修改起始id,出处,文件名
-starting_id = 22127 #起始id设置, 来自"寻找空闲题号"功能
+starting_id = 18315 #起始id设置, 来自"寻找空闲题号"功能
 raworigin = "" #题目来源的前缀(中缀在.tex文件中)
-filename = r"C:\Users\wangweiye\Documents\wwy sync\临时工作区\自拟题目15.tex" #题目的来源.tex文件
+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
 editor = "王伟叶" #编辑者姓名
-IndexDescription = "试题" #设置是否使用后缀, 留空("")则不用后缀, 不留空则以所设字符串作为后缀起始词, 按.tex文件中的顺序编号
+IndexDescription = "" #设置是否使用后缀, 留空("")则不用后缀, 不留空则以所设字符串作为后缀起始词, 按.tex文件中的顺序编号
 
 
 from database_tools import *
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index ef3aa1df..9808c474 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -472607,6 +472607,2386 @@
         "space": "4em",
         "unrelated": []
     },
+    "018315": {
+        "id": "018315",
+        "content": "判断下列各角分别属于哪个象限:\\\\\n(1) $-240^{\\circ}$;\\\\\n(2) $2100^{\\circ}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018316": {
+        "id": "018316",
+        "content": "写出与$-200^{\\circ}$的终边重合的所有角组成的集合$S$, 并列举$S$中满足不等式$-360^{\\circ} \\leq \\beta<720^{\\circ}$的所有元素$\\beta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018317": {
+        "id": "018317",
+        "content": "已知$\\alpha$为锐角, 且$\\alpha$的终边与$7 \\alpha$的终边关于$x$轴对称, 求$\\alpha$的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018318": {
+        "id": "018318",
+        "content": "已知集合$A=\\{\\alpha | 30^{\\circ}+k \\cdot 180^{\\circ}<\\alpha<90^{\\circ}+k \\cdot 180^{\\circ}, k \\in \\mathbf{Z}\\}$, 集合$B=\\{\\beta |-45^{\\circ}+k \\cdot 360^{\\circ}<\\beta<45^{\\circ}+k \\cdot 360^{\\circ}, k \\in \\mathbf{Z}\\}$, 求$A \\cap B$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018319": {
+        "id": "018319",
+        "content": "已知$\\alpha$与$\\beta$都是锐角, $\\alpha+\\beta$的终边与$-280^{\\circ}$的终边重合、$\\alpha-\\beta$的终边与$-670^{\\circ}$的终边重合, 求$\\alpha$与$\\beta$的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018320": {
+        "id": "018320",
+        "content": "按下列要求, 将$75^{\\circ}$换算成弧度:\\\\\n(1) 精确值;\\\\\n(2) 近似值. (结果精确到 $0.001$ )",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018321": {
+        "id": "018321",
+        "content": "将 $2.1$ 弧度换算成角度. (用度数表示, 结果保留两位小数)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018322": {
+        "id": "018322",
+        "content": "根据一些常用特殊角的角度与弧度的对应关系, 填写下表.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 角度 &$0^{\\circ}$&$30^{\\circ}$&$45^{\\circ}$&$60^{\\circ}$& & &$135^{\\circ}$& &$180^{\\circ}$&$270^{\\circ}$&$360^{\\circ} $\\\\\n\\hline 弧度 & & & & &$\\dfrac{\\pi}{2}$&$\\dfrac{2 \\pi}{3}$& &$\\dfrac{5 \\pi}{6}$&$\\pi$& &$2 \\pi$\\\\\n\\hline\n\\end{tabular}\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018323": {
+        "id": "018323",
+        "content": "将$32^{\\circ} 18'$换算为弧度.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018324": {
+        "id": "018324",
+        "content": "在弧度制的条件下, 证明下列关于扇形的公式:\\\\\n(1) $l=\\alpha r$;\\\\\n(2) $S=\\dfrac{1}{2} \\alpha r^2$;\\\\\n(3) $S=\\dfrac{1}{2} l r$.\\\\\n其中$r$是圆的半径, $\\alpha$($0<\\alpha<2 \\pi$)为圆心角, $l$是扇形的弧长, $S$是扇形的面积.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018325": {
+        "id": "018325",
+        "content": "写出终边在$x$轴上的所有角组成的集合. (用弧度制表示)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018326": {
+        "id": "018326",
+        "content": "设$\\alpha$是第二象限的角, 判断$\\dfrac{\\alpha}{2}$是哪个象限的角.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018327": {
+        "id": "018327",
+        "content": "已知扇形的圆心角为$2$弧度, 其圆心角所对弦长为$2$厘米, 则扇形面积是多少平方厘米? (结果精确到 $0.1$ 平方厘米)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018328": {
+        "id": "018328",
+        "content": "在直角$\\triangle POB$中 (如图所示), $\\angle PBO=90^{\\circ}$, 以$O$为圆心、$OB$为半径作圆弧交$OP$于点$A$. 若弧$AB$等分$\\triangle POB$的面积, 且$\\angle AOB=\\alpha$弧度, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$B$} coordinate (B);\n\\draw (2,0) node [below right] {$O$} coordinate (O);\n\\draw (0,3) node [above] {$P$} coordinate (P);\n\\draw (B) arc (180:{180-atan(3/2)}:2) node [above right] {$A$} coordinate (A);\n\\draw (P)--(B)--(O)--cycle;\n\\draw (B) pic [draw, scale = 0.3] {right angle = O--B--P};\n\\draw (O) pic [draw, scale = 0.3, \"$\\alpha$\", angle eccentricity = 2.5] {angle = P--O--B};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\tan \\alpha=\\alpha$}{$\\tan \\alpha=2 \\alpha$}{$\\sin \\alpha=2 \\cos \\alpha$}{$2 \\sin \\alpha=\\cos \\alpha$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018329": {
+        "id": "018329",
+        "content": "已知相互啮合的两个齿轮, 大轮有$48$齿, 小轮有$20$齿.\\\\\n(1) 当大轮转动一周时, 求小轮转动的角的大小;\\\\\n(2) 如果大轮的转速为$180$转/分, 小轮的半径为 $10.5$ 厘米, 那么小轮的圆周上一点每$1$秒转过的弧长是多少?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018330": {
+        "id": "018330",
+        "content": "已知角$\\alpha$的终边经过点$P(1,-2)$, 求角$\\alpha$的正弦、余弦、正切及余切值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018331": {
+        "id": "018331",
+        "content": "已知角$\\alpha$的终边经过点$P(a,-2 a)$($a<0$), 求角$\\alpha$的正弦、余弦、正切及余切值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018332": {
+        "id": "018332",
+        "content": "已知角$\\alpha$的终边经过点$P(-2,0)$, 求角$\\alpha$的正弦、余弦、正切及余切值. 填写下表.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline 角$\\alpha$& 0 &$\\dfrac{\\pi}{2}$&$\\pi$&$\\dfrac{3 \\pi}{2}$\\\\\n\\hline$\\sin \\alpha$& & & & \\\\\n\\hline$\\cos \\alpha$& & & & \\\\\n\\hline$\\tan \\alpha$& & & & \\\\\n\\hline$\\cot \\alpha$& & & & \\\\\n\\hline\n\\end{tabular}\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018333": {
+        "id": "018333",
+        "content": "若角$\\alpha$满足$\\sin \\alpha>0$, 且$\\tan \\alpha<0$, 则角$\\alpha$属于第几象限?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018334": {
+        "id": "018334",
+        "content": "已知角$\\alpha$的终边经过点$P(3 m,-4 m)$($m<0$), 求角$\\alpha$的正弦、正切值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018335": {
+        "id": "018335",
+        "content": "设$\\alpha$是三角形的一个内角, 在$\\sin \\alpha, \\cos \\alpha, \\tan \\alpha, \\cot \\dfrac{\\alpha}{2}$中, 有可能取负值的是\\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018336": {
+        "id": "018336",
+        "content": "已知角$\\alpha$的终边在直线$y=-3 x$上, 求$\\sin \\alpha, \\cos \\alpha, \\tan \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018337": {
+        "id": "018337",
+        "content": "已知$y=\\dfrac{\\sin \\alpha}{|\\sin \\alpha|}+\\dfrac{\\cos \\alpha}{|\\cos \\alpha|}+\\dfrac{\\tan \\alpha}{|\\tan \\alpha|}+\\dfrac{\\cot \\alpha}{|\\cot \\alpha|}$, 其中$\\alpha \\neq \\dfrac{k \\pi}{2}$($k \\in \\mathbf{Z}$), 请写出$y$所有可能的值.\n例1 求角$\\dfrac{5 \\pi}{4}$的正弦、余弦和正切值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018338": {
+        "id": "018338",
+        "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, 且$\\alpha$是第二象限的角.求$\\cos \\alpha, \\tan \\alpha$及$\\cot \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018339": {
+        "id": "018339",
+        "content": "已知$\\tan \\alpha=-\\dfrac{5}{12}$, 求$\\sin \\alpha$、$\\cos \\alpha$及$\\cot \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018340": {
+        "id": "018340",
+        "content": "已知$\\tan \\alpha=m$($m \\neq 0$), 且$\\alpha \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$, 用$m$表示$\\sin \\alpha$、$\\cos \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018341": {
+        "id": "018341",
+        "content": "已知$\\cos \\alpha=-\\dfrac{2 \\sqrt{2}}{3}$, 且$\\sin \\alpha>0$, 求$\\tan \\alpha+\\cot \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018342": {
+        "id": "018342",
+        "content": "已知$\\tan \\alpha=2$, 且$\\alpha \\in(2 \\pi, \\dfrac{5 \\pi}{2})$, 求$\\sin \\alpha+\\cos \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018343": {
+        "id": "018343",
+        "content": "已知$\\dfrac{4}{\\cos \\alpha}+\\tan \\alpha=8$, 求$\\sin \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018344": {
+        "id": "018344",
+        "content": "已知$\\sin \\theta=\\dfrac{1-a}{1+a}$, $\\cos \\theta=\\dfrac{3 a-1}{1+a}$, 且$\\theta \\in(\\dfrac{3 \\pi}{2}, \\dfrac{5 \\pi}{2})$, 求$\\theta$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018345": {
+        "id": "018345",
+        "content": "已知$\\tan \\alpha=m$, 用$m$表示$\\sin \\alpha$、$\\cos \\alpha$及$\\cot \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018346": {
+        "id": "018346",
+        "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 求$\\sin \\alpha \\cos \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018347": {
+        "id": "018347",
+        "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 且$\\alpha \\in(0, \\pi)$, 求$\\sin \\alpha-\\cos \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018348": {
+        "id": "018348",
+        "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{1}{5}$, 求$\\tan \\alpha+\\cot \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018349": {
+        "id": "018349",
+        "content": "已知$\\tan \\alpha=\\dfrac{1}{2}$, 求$\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha-\\cos ^2 \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018350": {
+        "id": "018350",
+        "content": "已知$\\tan \\alpha=\\dfrac{1}{2}$, 求$\\dfrac{\\sin ^2 \\alpha-\\sin \\alpha \\cos \\alpha}{\\sin ^2 \\alpha-2}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018351": {
+        "id": "018351",
+        "content": "证明下列恒等式:\\\\\n(1) $1+\\tan ^2 \\alpha=\\dfrac{1}{\\cos ^2 \\alpha}$;\\\\\n(2) $1+\\cot ^2 \\alpha=\\dfrac{1}{\\sin ^2 \\alpha}$;\\\\\n(3) $\\dfrac{1+\\cos \\alpha}{\\sin \\alpha}=\\dfrac{\\sin \\alpha}{1-\\cos \\alpha}$;\\\\\n(4) $\\dfrac{\\sin ^2 \\alpha-\\sin ^2 \\beta}{\\tan ^2 \\alpha-\\tan ^2 \\beta}=\\cos ^2 \\alpha \\cos ^2 \\beta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018352": {
+        "id": "018352",
+        "content": "已知$\\sin \\alpha+\\cos \\alpha=\\dfrac{7}{13}$, $\\alpha \\in(0, \\pi)$. 求$\\sin \\alpha$和$\\cos \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018353": {
+        "id": "018353",
+        "content": "已知$\\sin \\alpha-\\cos \\alpha=\\dfrac{7}{5}$, $\\alpha \\in(0, \\pi)$, 求$\\sin ^3 \\alpha+\\cos ^3 \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018354": {
+        "id": "018354",
+        "content": "证明: $\\dfrac{\\tan \\alpha \\cdot \\sin \\alpha}{\\tan \\alpha-\\sin \\alpha}=\\dfrac{\\tan \\alpha+\\sin \\alpha}{\\tan \\alpha \\cdot \\sin \\alpha}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018355": {
+        "id": "018355",
+        "content": "利用诱导公式求值:\\\\\n(1) $\\sin \\dfrac{20}{3} \\pi$;\\\\\n(2) $\\cos (-\\dfrac{7}{6} \\pi)$;\\\\\n(3) $\\tan (-\\dfrac{19}{4} \\pi)$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018356": {
+        "id": "018356",
+        "content": "化简: $\\dfrac{\\sin (2 \\pi-\\alpha) \\tan (\\pi+\\alpha) \\cot (-\\pi-\\alpha)}{\\cos (\\pi-\\alpha) \\tan (3 \\pi-\\alpha)}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018357": {
+        "id": "018357",
+        "content": "化简:\\\\\n(1) $\\dfrac{\\sin (-\\alpha)}{\\sin (\\pi+\\alpha)}-\\dfrac{\\cot (\\pi-\\alpha)}{\\cot (\\pi+\\alpha)}+\\dfrac{\\cos (2 \\pi-\\alpha)}{\\cos (3 \\pi-\\alpha)}$;\\\\\n(2) $\\dfrac{\\sin (\\pi-\\alpha)}{\\tan (4 \\pi+\\alpha)} \\cdot \\dfrac{\\tan (2 \\pi-\\alpha)}{\\cos (\\pi-\\alpha)} \\cdot \\dfrac{\\cos (-\\alpha)}{\\sin (\\pi+\\alpha)}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018358": {
+        "id": "018358",
+        "content": "求$\\cos 1^{\\circ}+\\cos 2^{\\circ}+\\cdots+\\cos 179^{\\circ}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018359": {
+        "id": "018359",
+        "content": "已知$\\alpha \\neq \\dfrac{n \\pi}{2}$, $n \\in \\mathbf{Z}$, 化简: $\\dfrac{\\sin (k \\pi-\\alpha)}{\\sin (k \\pi+\\alpha)}+\\dfrac{\\cos (k \\pi-\\alpha)}{\\cos (k \\pi+\\alpha)}+\\dfrac{\\tan (k \\pi-\\alpha)}{\\tan (k \\pi+\\alpha)}$, 其中$k \\in \\mathbf{Z}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018360": {
+        "id": "018360",
+        "content": "证明:\\\\\n(1) $\\sin (\\dfrac{3 \\pi}{2}+\\alpha)=-\\cos \\alpha$;\\\\\n(2) $\\cos (\\dfrac{3 \\pi}{2}+\\alpha)=\\sin \\alpha$;\\\\\n(3) $\\tan (\\dfrac{3 \\pi}{2}+\\alpha)=-\\cot \\alpha$;\\\\\n(4) $\\cot (\\dfrac{3 \\pi}{2}+\\alpha)=-\\tan \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018361": {
+        "id": "018361",
+        "content": "化简: $\\dfrac{\\sin (\\dfrac{\\pi}{2}+\\alpha) \\cos (\\dfrac{\\pi}{2}+\\alpha) \\sin (\\dfrac{\\pi}{2}-\\alpha)}{\\tan (\\dfrac{\\pi}{2}+\\alpha) \\cos (\\dfrac{3 \\pi}{2}+\\alpha) \\sin (-\\pi+\\alpha)}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018362": {
+        "id": "018362",
+        "content": "已知点$A$的坐标为$(-\\dfrac{3}{5}, \\dfrac{4}{5})$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{\\pi}{2}$至$OA'$. 求点$A'$的坐标.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018363": {
+        "id": "018363",
+        "content": "已知点$A$的坐标为$(1,1)$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{3 \\pi}{2}$至$OA'$. 求点$A'$的坐标.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018364": {
+        "id": "018364",
+        "content": "已知$\\alpha$为钝角, 且$\\sin (\\dfrac{\\pi}{4}+\\alpha)=\\dfrac{3}{4}$, 求$\\sin (\\dfrac{\\pi}{4}-\\alpha)$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018365": {
+        "id": "018365",
+        "content": "化简下列各式, 其中$k \\in \\mathbf{Z}$:\\\\\n(1) $\\sin (\\dfrac{k \\pi}{2}+\\alpha)$;\\\\\n(2) $\\cos (\\dfrac{k \\pi}{2}-\\alpha)$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018366": {
+        "id": "018366",
+        "content": "根据下列条件, 分别求角$x$:\\\\\n(1) 已知$\\sin x=\\dfrac{\\sqrt{3}}{2}$;\\\\\n(2) 已知$\\cos x=-\\dfrac{\\sqrt{2}}{2}$;\\\\\n(3) 已知$\\tan x=\\dfrac{\\sqrt{3}}{3}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018367": {
+        "id": "018367",
+        "content": "分别求满足下列条件的角$x$的集合:\n(1) $\\sin 2 x=\\dfrac{\\sqrt{3}}{2}$, $x \\in[0,2 \\pi]$;\\\\\n(2) $\\cos (x+\\dfrac{\\pi}{6})=\\dfrac{\\sqrt{2}}{2}$;\\\\\n(3) $\\tan (2 x+\\dfrac{\\pi}{3})=-\\dfrac{\\sqrt{3}}{3}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018368": {
+        "id": "018368",
+        "content": "已知角$x$满足$\\sin x=\\dfrac{\\sqrt{2}}{2}$.\\\\\n(1) 若$x \\in(0, \\dfrac{\\pi}{2})$, 则$x=$\\blank{50};\\\\\n(2) 若$x \\in(0, \\pi)$, 则$x=$\\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018369": {
+        "id": "018369",
+        "content": "根据下列条件, 分别求角$x$:\\\\\n(1) $\\sin ^2 x-2 \\sin x-3=0$;\\\\\n(2) $2 \\sqrt{3} \\sin ^2 x=\\cos x$;\\\\\n(3) $\\sin x=\\cos x$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018370": {
+        "id": "018370",
+        "content": "满足$\\cos (\\pi \\cos x)=0$, $x \\in[0, \\dfrac{3 \\pi}{2}]$的角$x$组成的集合为\\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018371": {
+        "id": "018371",
+        "content": "满足$\\sin 5 x=\\cos x, x \\in(0, \\pi)$的角$x$的个数为\\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018372": {
+        "id": "018372",
+        "content": "利用两角和与差的余弦公式, 求$\\cos 75^{\\circ}$和$\\cos 15^{\\circ}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018373": {
+        "id": "018373",
+        "content": "已知$\\sin \\alpha=\\dfrac{3}{5}$, $\\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$, $\\cos \\beta=\\dfrac{5}{13}$, $\\beta \\in(\\dfrac{3 \\pi}{2}, 2 \\pi)$, 求$\\cos (\\alpha-\\beta)$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018374": {
+        "id": "018374",
+        "content": "化简:\\\\\n(1) $\\cos 72^{\\circ} \\cos 18^{\\circ}-\\sin 72^{\\circ} \\sin 18^{\\circ}$;\\\\\n(2) $\\cos (\\alpha-\\dfrac{\\pi}{4}) \\cos \\alpha+\\sin (\\alpha-\\dfrac{\\pi}{4}) \\sin \\alpha$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018375": {
+        "id": "018375",
+        "content": "若$\\alpha$、$\\beta$为锐角, $\\sin \\alpha=\\dfrac{4 \\sqrt{3}}{7}$, $\\cos (\\alpha+\\beta)=-\\dfrac{11}{14}$, 求角$\\beta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018376": {
+        "id": "018376",
+        "content": "已知$\\cos (\\theta-\\dfrac{\\pi}{4})=-\\dfrac{\\sqrt{2}}{10}$, 且$\\theta$是第二象限的角. 求$\\cos \\theta$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018377": {
+        "id": "018377",
+        "content": "化简:\\\\\n(1) $\\cos ^215^{\\circ}-\\sin ^215^{\\circ}$;\\\\\n(2) $\\cos ^2 \\alpha-\\sin ^2 \\alpha$;\\\\\n(3) $\\sin ^2 \\dfrac{3}{2} x-\\cos ^2 \\dfrac{3}{2} x$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018378": {
+        "id": "018378",
+        "content": "利用两角差的正弦公式, 求$\\sin 15^{\\circ}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018379": {
+        "id": "018379",
+        "content": "证明: $\\sin (\\alpha+\\beta) \\sin (\\alpha-\\beta)=\\sin ^2 \\alpha-\\sin ^2 \\beta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018380": {
+        "id": "018380",
+        "content": "已知$\\tan \\alpha=\\dfrac{1}{3}$, $\\tan \\beta=-2$. 求:\\\\\n(1) $\\tan (\\alpha+\\beta)$;\\\\\n(2) $\\cot (\\alpha-\\beta)$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018381": {
+        "id": "018381",
+        "content": "利用两角和的正切公式, 求$\\dfrac{1+\\tan 75^{\\circ}}{1-\\tan 75^{\\circ}}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018382": {
+        "id": "018382",
+        "content": "已知$\\tan \\alpha, \\tan \\beta$是方程$4 x^2-3 \\sqrt{3} x+1=0$的两根, 若$\\alpha, \\beta \\in(-\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})$, 求$\\alpha+\\beta$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018383": {
+        "id": "018383",
+        "content": "求证: $\\tan 1^{\\circ}$不是有理数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018384": {
+        "id": "018384",
+        "content": "不用计算器, 求$\\tan 20^{\\circ}+\\tan 40^{\\circ}+\\sqrt{3} \\tan 20^{\\circ} \\tan 40^{\\circ}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018385": {
+        "id": "018385",
+        "content": "若$\\triangle ABC$不是直角三角形, 求证: $\\tan A+\\tan B+\\tan C=\\tan A \\tan B \\tan C$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018386": {
+        "id": "018386",
+        "content": "如图, 已知点$A$的坐标为$(1,2)$, 将$OA$绕坐标原点$O$逆时针旋转$\\dfrac{\\pi}{4}$至$OA'$. 求点$A'$的坐标.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw [->] (-3, 0) -- (3, 0) node [below] {$x$};\n\\draw [->] (0, -1) -- (0, 3) node [left] {$y$};\n\\draw (0, 0) node [below left] {$O$} coordinate (O);\n\\filldraw (1, 2) circle (0.05) node [right] {$A$} coordinate (A);\n\\filldraw ($(O)!1!45: (A)$) circle (0.05) node [left] {$A'$} coordinate (A');\n\\draw (A)--(O)--(A');\n\\draw (O) pic [draw, ->, scale = 0.6, \"$\\alpha$\", angle eccentricity = 1.8] {angle = A--O--A'};\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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018387": {
+        "id": "018387",
+        "content": "把下列各式化为$A \\sin (\\alpha+\\varphi)$($A>0$)的形式:\\\\\n(1) $\\dfrac{1}{2} \\sin \\alpha+\\dfrac{\\sqrt{3}}{2} \\cos \\alpha$;\\\\\n(2) $\\sin \\alpha-\\cos \\alpha$;\\\\\n(3) $3 \\sin \\alpha+4 \\cos \\alpha$;\\\\\n(4) $a \\sin \\alpha+b \\cos \\alpha$($a b \\neq 0$).",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018388": {
+        "id": "018388",
+        "content": "若存在角$\\alpha$, 使得$\\sqrt{3} \\sin \\alpha+\\cos \\alpha=4-m$, 求实数$m$的取值范围.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018389": {
+        "id": "018389",
+        "content": "$A, B, C$为$\\triangle ABC$的内角, $\\triangle ABC$不为直角三角形. 若$\\sqrt{3} \\tan C-1=\\dfrac{\\tan B+\\tan C}{\\tan A}$, 求角$B$的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018390": {
+        "id": "018390",
+        "content": "已知角$\\alpha$的终边经过点$A(\\dfrac{\\sqrt{3}}{2}, \\dfrac{1}{2})$, 将$OA$绕坐标原点$O$顺时针旋转$\\dfrac{3 \\pi}{4}$至$OB$, 若角$\\beta$的终边经过点$B$, 求$\\cos (\\alpha+\\beta)$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018391": {
+        "id": "018391",
+        "content": "已知$\\alpha$、$\\beta$为任意角, ``$\\sin \\alpha \\sin \\beta+\\cos \\alpha \\cos \\beta=0$''是``$\\sin \\alpha \\cos \\beta-\\cos \\alpha \\sin \\beta=1$''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018392": {
+        "id": "018392",
+        "content": "已知角$\\alpha$的终边经过点$P(1-\\tan \\dfrac{\\pi}{12}, 1+\\tan \\dfrac{\\pi}{12})$, 且$0<\\alpha<2 \\pi$, 求角$\\alpha$的大小.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018393": {
+        "id": "018393",
+        "content": "已知$\\sin \\alpha=\\dfrac{4}{5}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$. 求$\\sin 2 \\alpha$、$\\cos 2 \\alpha$和$\\tan 2 \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018394": {
+        "id": "018394",
+        "content": "已知$\\sin \\alpha=\\dfrac{4}{5}, \\alpha \\in(\\dfrac{\\pi}{2}, \\pi)$. 求$\\sin 4 \\alpha$、$\\cos 4 \\alpha$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018395": {
+        "id": "018395",
+        "content": "试用$\\cos \\theta$表示$\\cos 3 \\theta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018396": {
+        "id": "018396",
+        "content": "证明:\\\\\n(1) $2 \\cos ^2 \\theta+2 \\sin \\theta \\cos \\theta-1=\\sqrt{2} \\sin (2 \\theta+\\dfrac{\\pi}{4})$;\\\\\n(2) $\\dfrac{1+\\sin 2 \\theta+\\cos 2 \\theta}{1+\\sin 2 \\theta-\\cos 2 \\theta}=\\cot \\theta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018397": {
+        "id": "018397",
+        "content": "若$\\sin \\theta=\\dfrac{3}{5}, \\cos \\theta=-\\dfrac{4}{5}$, 则角$2 \\theta$的终边在第\\blank{50}象限.",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018398": {
+        "id": "018398",
+        "content": "已知$\\sin (\\dfrac{\\pi}{4}-x)=\\dfrac{3}{5}$, 求$\\sin 2 x$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018399": {
+        "id": "018399",
+        "content": "已知$\\alpha \\in(\\dfrac{3 \\pi}{2}, \\dfrac{5 \\pi}{2})$, 则$\\sqrt{\\dfrac{1}{2}+\\dfrac{1}{2} \\cos \\alpha}$等于\\bracket{20}.\n\\fourch{$\\sin \\dfrac{\\alpha}{2}$}{$\\cos \\dfrac{\\alpha}{2}$}{$-\\sin \\dfrac{\\alpha}{2}$}{$-\\cos \\dfrac{\\alpha}{2}$}",
+        "objs": [],
+        "tags": [],
+        "genre": "选择题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018400": {
+        "id": "018400",
+        "content": "如图是来自古希腊数学家希波克拉底所研究的几何图形, 此图由三个半圆构成, 三个半圆的直径分别为直角三角形$ABC$的斜边$BC$, 直角边$AB, AC$. 已知以直角边$AC, AB$为直径的半圆的面积之比为$\\dfrac{1}{4}$, 记$\\angle ABC=\\alpha$, 则$\\sin 2 \\alpha=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.5]\n\\draw (0, 0) node [below] {$B$} coordinate (B);\n\\draw (5, 0) node [below] {$C$} coordinate (C);\n\\draw (4, 2) node [above] {$A$} coordinate (A);\n\\draw (C) arc (0: 180: 2.5);\n\\draw (C) arc ({-atan(2)}: {180-atan(2)}: {sqrt(5)/2});\n\\draw (B) arc ({180+atan(0.5)}: {atan(0.5)}: {sqrt(5)});\n\\draw (A)--(B)--(C)--cycle;\n\\end{tikzpicture}\n\\end{center}",
+        "objs": [],
+        "tags": [],
+        "genre": "填空题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "",
+        "unrelated": []
+    },
+    "018401": {
+        "id": "018401",
+        "content": "用$\\cos \\alpha$分别表示$\\cos ^2 \\dfrac{\\alpha}{2}, \\sin ^2 \\dfrac{\\alpha}{2}$及$\\tan ^2 \\dfrac{\\alpha}{2}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018402": {
+        "id": "018402",
+        "content": "证明: $\\tan \\dfrac{\\alpha}{2}=\\dfrac{\\sin \\alpha}{1+\\cos \\alpha}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018403": {
+        "id": "018403",
+        "content": "证明: $\\sin \\alpha \\cos \\beta=\\dfrac{1}{2}[\\sin (\\alpha+\\beta)+\\sin (\\alpha-\\beta)]$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018404": {
+        "id": "018404",
+        "content": "证明: $\\sin \\alpha+\\sin \\beta=2 \\sin \\dfrac{\\alpha+\\beta}{2} \\cos \\dfrac{\\alpha-\\beta}{2}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018405": {
+        "id": "018405",
+        "content": "证明下列恒等式:\\\\\n(1) $\\sin \\alpha=\\dfrac{2 \\tan \\dfrac{\\alpha}{2}}{1+\\tan ^2 \\dfrac{\\alpha}{2}}$;\\\\\n(2) $\\cos \\alpha=\\dfrac{1-\\tan ^2 \\dfrac{\\alpha}{2}}{1+\\tan ^2 \\dfrac{\\alpha}{2}}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018406": {
+        "id": "018406",
+        "content": "证明下列恒等式:\\\\\n(1) $\\dfrac{1-\\cos 4 \\alpha}{\\sin 4 \\alpha} \\cdot \\dfrac{\\cos 2 \\alpha}{1+\\cos 2 \\alpha}=\\tan \\alpha$;\\\\\n(2) $\\tan \\dfrac{3 \\alpha}{2}-\\tan \\dfrac{\\alpha}{2}=\\dfrac{2 \\sin \\alpha}{\\cos \\alpha+\\cos 2 \\alpha}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018407": {
+        "id": "018407",
+        "content": "已知$2 \\sin \\alpha=\\sin \\theta+\\cos \\theta$, $\\sin ^2 \\beta=\\sin \\theta \\cos \\theta$, 求证: $2 \\cos 2 \\alpha=\\cos 2 \\beta$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018408": {
+        "id": "018408",
+        "content": "某同学在一次研究性学习中发现, 以下五个式子的值都等于同一个常数:\\\\\n\\textcircled{1} $\\sin ^213^{\\circ}+\\cos ^217^{\\circ}-\\sin 13^{\\circ} \\cos 17^{\\circ}$;\n\\textcircled{2} $\\sin ^215^{\\circ}+\\cos ^215^{\\circ}-\\sin 15^{\\circ} \\cos 15^{\\circ}$;\n\\textcircled{3} $\\sin ^218^{\\circ}+\\cos ^212^{\\circ}-\\sin 18^{\\circ} \\cos 12^{\\circ}$;\n\\textcircled{4} $\\sin ^2(-18^{\\circ})+\\cos ^248^{\\circ}-\\sin (-18^{\\circ}) \\cos 48^{\\circ}$;\n\\textcircled{5} $\\sin ^2(-25^{\\circ})+\\cos ^255^{\\circ}-\\sin (-25^{\\circ}) \\cos 55^{\\circ}$.\\\\\n(1) 试从上述五个式子中选择一个, 求出这个常数;\\\\\n(2) 根据 (1) 的计算结果, 将该同学的发现推广为三角恒等式, 并证明你的结论.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018409": {
+        "id": "018409",
+        "content": "在$\\triangle ABC$中, 已知$\\angle CAB=130^{\\circ}$, $\\angle CBA=30^{\\circ}$, $AB=10 \\text{km}$. 求$AC$与$BC$的长. (结果精确到$0.1 \\text{km}$)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018410": {
+        "id": "018410",
+        "content": "已知圆$O$是$\\triangle ABC$的外接圆, 其圆心为$O$, 直径为$2R$. 试用$R$与角$A$、$B$及$C$的正弦来表示三角形的边长$a$、$b$及$c$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018411": {
+        "id": "018411",
+        "content": "设$R$是$\\triangle ABC$的外接圆的半径, $S$为$\\triangle ABC$的面积. 求证:\\\\\n(1) $S=\\dfrac{a b c}{4R}$;\\\\\n(2) $S=2R^2 \\sin A \\sin B \\sin C$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018412": {
+        "id": "018412",
+        "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对边的边长分别记作$a$、$b$、$c$. 若$b=5, B=\\dfrac{\\pi}{4}$, $\\sin A=\\dfrac{1}{3}$, 求$c$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018413": {
+        "id": "018413",
+        "content": "在$\\triangle ABC$中, 角$A$、$B$、$C$所对边的边长分别记作$a$、$b$、$c$. 已知$A=\\dfrac{\\pi}{4}$, $b \\sin (\\dfrac{\\pi}{4}+C)-c \\sin (\\dfrac{\\pi}{4}+B)=a$.\\\\\n(1) 求$B-C$;\\\\\n(2) 若$a=\\sqrt{2}$, 求$\\triangle ABC$的面积.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018414": {
+        "id": "018414",
+        "content": "在$\\triangle ABC$中, 已知$a=\\sqrt{6}$, $b=\\sqrt{3}+1$, $C=45^{\\circ}$. 求$c$、$A$及$B$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018415": {
+        "id": "018415",
+        "content": "在$\\triangle ABC$中, 已知$a=2$, $b=2 \\sqrt{3}$, $A=30^{\\circ}$, 求$B$、$C$及$c$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018416": {
+        "id": "018416",
+        "content": "在$\\triangle ABC$中, 已知$a=4$, $b=5$, $c=6$. 求角$A$的余弦值和$\\triangle ABC$的面积$S$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018417": {
+        "id": "018417",
+        "content": "已知三角形的三边长是三个连续的正整数.\\\\\n(1) 若此三角形是钝角三角形, 求三边的长;\\\\\n(2) 若此三角形最大角是最小角的两倍, 求三边的长.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018418": {
+        "id": "018418",
+        "content": "在$\\triangle ABC$中, 已知$b^2+c^2-b c=a^2$, 且$\\dfrac{b}{c}=\\dfrac{\\tan B}{\\tan C}$. 求证: $\\triangle ABC$为等边三角形.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018419": {
+        "id": "018419",
+        "content": "在$\\triangle ABC$中, 已知$b^2+c^2-b c=a^2$, 且$\\dfrac{b}{c}=\\dfrac{\\cos B}{\\cos C}$, 请判断$\\triangle ABC$的形状.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018420": {
+        "id": "018420",
+        "content": "在$\\triangle ABC$中, 已知$a=5$, $b=4$, 三角形的面积$S=8$. 求$c$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018421": {
+        "id": "018421",
+        "content": "根据下列条件, 分别求角$x$:\\\\\n(1) 已知$\\sin x=\\dfrac{1}{3}$;\\\\\n(2) 已知$\\cos x=-\\dfrac{3}{5}$, $x \\in[0, \\pi]$;\\\\\n(3) 已知$\\tan x=-3$, $x \\in(\\dfrac{\\pi}{2}, \\dfrac{3 \\pi}{2})$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018422": {
+        "id": "018422",
+        "content": "已知$\\cos 2 \\alpha=\\dfrac{7}{25}$, $\\alpha \\in(0, \\dfrac{\\pi}{2})$, $\\sin \\beta=-\\dfrac{5}{13}$, $\\beta \\in(\\pi, \\dfrac{3 \\pi}{2})$, 求$\\alpha+\\beta$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018423": {
+        "id": "018423",
+        "content": "在$\\triangle ABC$中, 已知$a \\sin A+b(\\sin A+\\sin B)-c \\sin C=0$.\\\\ \n(1) 求$C$;\\\\\n(2) 若$c=2$, 求$a+b$的最大值.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018424": {
+        "id": "018424",
+        "content": "在$\\triangle ABC$中, 已知$\\dfrac{a}{\\cos A}=\\dfrac{b}{\\cos B}=\\dfrac{c}{\\cos C}$, 判断这个三角形的形状.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018425": {
+        "id": "018425",
+        "content": "金茂大厦是改革开放以来上海出现的超高层标志性建筑. 有一位测量爱好者在与金茂大厦底部同一水平线上的$B$处测得金茂大厦顶部$A$的仰角为$15.66^{\\circ}$, 再向金茂大厦前进$500 \\mathrm{m}$到达$C$处, 测得金茂大厦顶部$A$的仰角为$22.81^{\\circ}$. 请根据以上数据估算出金茂大厦的高度. (结果精确到$1 \\mathrm{m}$)",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018426": {
+        "id": "018426",
+        "content": "甲船在距离$A$港口$24$海里并在南偏西$20^{\\circ}$方向的$C$处驻留等候进港, 乙船在$A$港口南偏东$40^{\\circ}$方向的$B$处沿直线行驶入港, 甲、乙两船距离为$31$海里. 当乙船行驶$20$海里到达$D$处时, 接到港口指令, 前往救援忽然发生火灾的甲船. 求此时甲、乙两船之间的距离.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018427": {
+        "id": "018427",
+        "content": "甲船在距离$A$港口$24$海里并在南偏西$20^{\\circ}$方向的$C$处驻留等候进港, 乙船在$A$港口南偏东$40^{\\circ}$方向的$B$处沿直线行驶入港, 甲、乙两船距离为$31$海里. 当乙船行驶$20$海里到达$D$处时, 接到港口指令, 前往救援忽然发生火灾的甲船. 当乙船接到命令前往救援时, 应按何方向行驶可最快抵达甲船处?",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018428": {
+        "id": "018428",
+        "content": "如图, 有两条相交成$60^{\\circ}$角的直路$l_1, l_2$, 交点是$O$, 警务岗$A$、$B$分别在$l_1, l_2$上, 警务岗$A$离$O$点$1$千米, 警务岗$B$离$O$点$3$千米. 若警员甲、乙分别从$A$、$B$同时出发, 甲沿$AC$方向, 乙沿$BO$方向, 均以$4$千米/小时的速度沿途巡逻. \n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-1, 0) -- (4, 0) node [below] {$l_2$};\n\\draw (60: -1)-- (60: 2.5) node [right] {$l_1$};\n\\filldraw (0, 0)  circle (0.03) node [below] {$O$} coordinate (O);\n\\filldraw (3, 0) circle (0.03) node [below] {$B$} coordinate (B);\n\\filldraw (60: {8/(3+sqrt(3))}) circle (0.03) node [above left] {$C$} coordinate (C);\n\\filldraw (60: 1) circle (0.03) node [above left] {$A$} coordinate (A);\n\\draw (O) pic [draw, \"$60^\\circ$\", scale = 0.5, angle eccentricity = 2.5] {angle = B--O--A};\n\\end{tikzpicture}\n\\end{center}\n(1) 当警员甲行至点$C$处时, $\\angle OBC=45^{\\circ}$, 求$OC$之间的距离;\\\\\n(2) 设$t$小时后甲、乙两人之间的距离为$S$千米, 试将$S$表示为$t$的函数.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018429": {
+        "id": "018429",
+        "content": "化简$\\dfrac{\\sin (\\theta-5 \\pi)}{\\tan (3 \\pi-\\theta)} \\cdot \\dfrac{\\cot (\\dfrac{\\pi}{2}-\\theta)}{\\tan (\\theta-\\dfrac{3 \\pi}{2})} \\cdot \\dfrac{\\cos (8 \\pi-\\theta)}{\\cos (\\theta+\\dfrac{\\pi}{2})}$.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018430": {
+        "id": "018430",
+        "content": "若$\\cos (\\dfrac{\\pi}{4}+x)=\\dfrac{3}{5}, \\pi<x<\\dfrac{7 \\pi}{4}$.\\\\\n(1) 求$\\cos x$的值;\\\\\n(2) 求$\\dfrac{\\sin 2 x+2 \\sin ^2 x}{1-\\tan x}$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018431": {
+        "id": "018431",
+        "content": "如图, 点$A$、$B$、$C$分别表示三座岛屿, 点$C$在点$A$的北偏东$47^{\\circ}$方向, 点$B$在点$C$的南偏西$36^{\\circ}$方向, 点$B$在点$A$的南偏东$79^{\\circ}$方向, 且$A$、$B$两点的距离为$3$海里.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.2]\n\\draw (0, 0) node [above left] {$A$} coordinate (A);\n\\draw (-11: 3) node [below] {$B$} coordinate (B);\n\\draw (43: {3/sin(11)*sin(115)}) node [above right] {$C$} coordinate (C);\n\\foreach \\i in {A, C}\n{\\draw [dashed] (\\i) ++ (-2, 0) --++ (4, 0) ++ (-2, -2) --++ (0, 4) node [above] {北};};\n\\draw (C) ++ (10, 0) node [above] {$P$} coordinate (P);\n\\draw (A)--(B)--(C)--cycle(C)--(P);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$A$、$C$两点间的距离; (结果精确到 $0.01$ 海里)\\\\\n(2) 某时刻, 我国一渔船在点$A$处因故障抛针发出求救信号.一艘 R 国舰艇正从点$C$正东$10$海里的点$P$处以$18$海里/小时的速度接近渔船, 其航线为$P \\to C \\to A$(沿直线行进), 而我国的救援船位于点$A$南偏西$60^{\\circ}$方向$20$海里的点$Q$处, 收到信号后赶往救助, 其航线为先向正北航行$8$海里至点$M$处, 再折向点$A$沿直线航行, 航速为$22$海里/小时. 问救援船能否先于 R 国舰艇赶到进行救助? 请说明理由.",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018432": {
+        "id": "018432",
+        "content": "至少用两种方法求$\\sin ^2 \\alpha \\cdot \\sin ^2 \\beta+\\cos ^2 \\alpha \\cdot \\cos ^2 \\beta-\\dfrac{1}{2} \\cos 2 \\alpha \\cos 2 \\beta$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例题与习题",
+        "edit": [
+            "20230705\t王伟叶"
+        ],
+        "same": [],
+        "related": [],
+        "remark": "",
+        "space": "4em",
+        "unrelated": []
+    },
+    "018433": {
+        "id": "018433",
+        "content": "如图, 某段海岸线可近似看作一条曲线, 该曲线由线段$AB$和四分之一圆弧$\\overset\\frown{BC}$构成, $D$为一海岛, $B$在$D$的正北方向, 且$B$、$D$相距 $39.2$ 千米, $A$在$D$的北偏西$58^{\\circ}$方向, $C$在$D$的北偏东$22^{\\circ}$方向, $C$在$B$的南偏东$68^{\\circ}$方向.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.06]\n\\draw (0, 0) node [above] {$B$} coordinate (B);\n\\draw (B) ++ (0, -39.2) node [below] {$D$} coordinate (D);\n\\draw (B) ++ (-22: {39.2*sin(22)}) node [right] {$C$} coordinate (C);\n\\draw (B) ++ ({270-(180-58-asin(39.2/40*sin(58)))}: 40) node [left] {$A$} coordinate (A);\n\\draw (A)-- (B) arc (113: 23: {39.2/2*sin(22)*sqrt(2)});\n\\draw [dashed] (A)--(D)--(C)--(B)--(D);\n\\draw pic [draw, scale = 0.9, \"$22^\\circ$\", angle eccentricity = 2.5] {angle = C--D--B};\n\\draw pic [draw, scale = 0.6, \"$58^\\circ$\", angle eccentricity = 1.8] {angle = B--D--A};\n\\draw pic [draw, scale = 0.6, \"$68^\\circ$\", angle eccentricity = 1.8] {angle = D--B--C};\n\\end{tikzpicture}\n\\end{center}\n(1) 若沿$\\overset\\frown{BC}$修建观光道, 计算该观光道$\\overset\\frown{BC}$的长度(精确到 $0.001$ 千米);\\\\\n(2) 现规划在该海岸线上选取一处$E$, 修建从$E$直通$D$的公路桥. 已知$A$、$B$相距$40$千米, 求公路桥$DE$的最短长度 (精确到 $0.001$ 千米).",
+        "objs": [],
+        "tags": [],
+        "genre": "解答题",
+        "ans": "",
+        "solution": "",
+        "duration": -1,
+        "usages": [],
+        "origin": "空中课堂必修第二册三角例题与习题",
+        "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.",