diff --git a/工具/批量添加题库字段数据.ipynb b/工具/批量添加题库字段数据.ipynb
index 0134b1d9..69dd8677 100644
--- a/工具/批量添加题库字段数据.ipynb
+++ b/工具/批量添加题库字段数据.ipynb
@@ -2,9 +2,100 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "题号: 004548 , 字段: objs 中已添加数据: K0309001B\n",
+      "题号: 003092 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 003094 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 003101 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 003102 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006100 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006115 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006116 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006117 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006120 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006123 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006125 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006126 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006128 , 字段: objs 中已添加数据: K0309003B\n",
+      "题号: 006138 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006145 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003095 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003096 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003097 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003098 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003099 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003100 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003103 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003104 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003105 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003107 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003108 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006101 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006104 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006124 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006129 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006146 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006147 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006148 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006149 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006152 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006153 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006154 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006155 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006156 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006157 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006158 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006159 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006162 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006163 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006164 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006165 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006166 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006167 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006168 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006169 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006170 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006171 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006172 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006173 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006174 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006175 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006176 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006191 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006232 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 006257 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 008176 , 字段: objs 中已添加数据: K0310002B\n",
+      "题号: 003093 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 004681 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 005086 , 字段: objs 中已有该数据: K0311002B\n",
+      "题号: 006177 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 008169 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 008170 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 010251 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 009572 , 字段: objs 中已添加数据: K0311002B\n",
+      "题号: 003112 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003113 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003114 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003115 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003117 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003125 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003126 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 003127 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 006190 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 006214 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 008385 , 字段: objs 中已添加数据: K0312003B\n",
+      "题号: 008168 , 字段: objs 中已添加数据: K0313002B\n",
+      "题号: 006286 , 字段: objs 中已添加数据: K0313003B\n",
+      "题号: 006288 , 字段: objs 中已添加数据: K0313003B\n"
+     ]
+    }
+   ],
    "source": [
     "import os,re,json\n",
     "\n",
@@ -88,6 +179,13 @@
     "with open(r\"../题库0.3/Problems.json\",\"w\",encoding = \"utf8\") as f:\n",
     "    f.write(json.dumps(pro_dict,indent=4,ensure_ascii=False))"
    ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
   }
  ],
  "metadata": {
diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json
index 58eda4b3..cfb8e6dd 100644
--- a/题库0.3/Problems.json
+++ b/题库0.3/Problems.json
@@ -76225,7 +76225,9 @@
     "003092": {
         "id": "003092",
         "content": "求值: $\\cos(31^\\circ-\\alpha)\\cos(29^\\circ+\\alpha)-\\sin(31^\\circ-\\alpha)\\sin(29^\\circ+\\alpha)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76246,7 +76248,9 @@
     "003093": {
         "id": "003093",
         "content": "将$\\sin\\alpha-\\sqrt 3\\cos\\alpha$化为$A\\sin(\\alpha+\\varphi)$的形式($A>0$, $\\varphi\\in [0,2\\pi)$): $\\sin\\alpha-\\sqrt 3\\cos\\alpha=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76267,7 +76271,9 @@
     "003094": {
         "id": "003094",
         "content": "若$\\sin \\alpha =\\dfrac 78$, $\\cos \\beta =-\\dfrac 14$, $\\alpha,\\beta$在同一象限, 则$\\cos(\\alpha-\\beta)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76288,7 +76294,9 @@
     "003095": {
         "id": "003095",
         "content": "已知$\\cos\\theta=-\\dfrac 35$, $\\theta\\in (\\dfrac{\\pi}2,\\pi)$, 则$\\sin(\\theta+\\dfrac{\\pi}4)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76309,7 +76317,9 @@
     "003096": {
         "id": "003096",
         "content": "若$\\alpha$为锐角, 且$\\sin(\\alpha-\\dfrac{\\pi}6)=\\dfrac 16$, 则$\\sin\\alpha=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76330,7 +76340,9 @@
     "003097": {
         "id": "003097",
         "content": "已知$\\tan(\\alpha+\\beta)=\\dfrac 23$, $\\tan(\\beta-\\dfrac{\\pi}4)=\\dfrac 14$, 则$\\tan(\\alpha+\\dfrac{\\pi}4)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76353,7 +76365,9 @@
     "003098": {
         "id": "003098",
         "content": "若$\\tan\\alpha$与$\\tan\\beta$是方程$3x^2+5x-2=0$的两个根, 且$0<\\alpha<\\dfrac{\\pi}2$, $\\dfrac{\\pi}2<\\beta<\\pi$, 则$\\alpha+\\beta$的值为\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76374,7 +76388,9 @@
     "003099": {
         "id": "003099",
         "content": "设$\\alpha,\\alpha+\\beta$均为象限角. 若$2\\sin\\beta=\\sin(2\\alpha+\\beta)$, 求$\\dfrac{\\tan(\\alpha+\\beta)}{\\tan\\alpha}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76395,7 +76411,9 @@
     "003100": {
         "id": "003100",
         "content": "*已知$\\tan\\alpha=-\\dfrac 17$, $\\tan\\beta=-\\dfrac 13$, 且$\\alpha,\\beta$均为钝角, 求$\\alpha+2\\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76416,7 +76434,9 @@
     "003101": {
         "id": "003101",
         "content": "*是否存在锐角$\\alpha ,\\beta ,\\theta$, 使得$\\sin\\theta=\\sin\\beta-\\sin\\alpha$, $\\cos\\theta=\\cos\\alpha-\\cos\\beta$? 若存在, 求出$\\alpha-\\beta$的所有可能值; 若不存在, 说明理由.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76437,7 +76457,9 @@
     "003102": {
         "id": "003102",
         "content": "若$\\sin\\alpha-\\sin\\beta=-\\dfrac 13$, $\\cos\\alpha-\\cos\\beta=\\dfrac 12$, 则$\\cos(\\alpha-\\beta)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76461,7 +76483,9 @@
     "003103": {
         "id": "003103",
         "content": "若$\\dfrac{\\pi}2<\\beta<\\alpha<\\dfrac{3\\pi}4$, $\\cos(\\alpha-\\beta)=\\dfrac{12}{13}$, $\\sin(\\alpha+\\beta)=-\\dfrac 35$, 则$\\sin2\\alpha=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76482,7 +76506,9 @@
     "003104": {
         "id": "003104",
         "content": "若$\\sin(\\alpha+\\beta)=\\dfrac 12$, $\\sin(\\alpha-\\beta)=\\dfrac 13$, 则$\\dfrac{\\tan\\alpha}{\\tan\\beta}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76506,7 +76532,9 @@
     "003105": {
         "id": "003105",
         "content": "若$\\sin A=\\dfrac{\\sqrt 5}5$, $\\sin B=\\dfrac{\\sqrt{10}}{10}$, 且$A,B$均为钝角, 则$A+B=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76548,7 +76576,9 @@
     "003107": {
         "id": "003107",
         "content": "设常数$m\\ne 0$, 若关于$x$的方程$mx^2+(2m-3)x+m-2=0$的两实数根为$\\tan\\alpha,\\tan\\beta$, 求$\\tan(\\alpha+\\beta)$的取值范围.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76569,7 +76599,9 @@
     "003108": {
         "id": "003108",
         "content": "是否存在锐角$\\alpha,\\beta$, 使得$\\alpha+2\\beta=\\dfrac{2\\pi}3$, 且$\\tan\\beta=(2-\\sqrt 3)\\cot\\dfrac{\\alpha}2$? 若存在, 求出所有的$\\alpha,\\beta$的值; 若不存在, 说明理由.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76655,7 +76687,9 @@
     "003112": {
         "id": "003112",
         "content": "若$\\tan\\theta=2$, 则$3\\cos 2\\theta+4\\sin 2\\theta=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76676,7 +76710,9 @@
     "003113": {
         "id": "003113",
         "content": "若$\\cos(\\alpha+\\dfrac{\\pi}4)=\\dfrac 35$, $\\dfrac{\\pi}2<\\alpha<\\dfrac{3\\pi}2$, 则$\\cos 2\\alpha=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76697,7 +76733,9 @@
     "003114": {
         "id": "003114",
         "content": "化简: $\\dfrac{\\tan (45^\\circ-\\alpha)}{1-\\tan^2(45^\\circ-\\alpha)}\\cdot \\dfrac{\\sin \\alpha \\cos \\alpha}{\\cos^2\\alpha -\\sin ^2\\alpha}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76720,7 +76758,9 @@
     "003115": {
         "id": "003115",
         "content": "若$\\tan\\dfrac{\\alpha}2+\\cot\\dfrac{\\alpha}2=\\dfrac 52$, 则$\\sin\\alpha=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76762,7 +76802,9 @@
     "003117": {
         "id": "003117",
         "content": "化简: $\\dfrac{2\\tan (\\dfrac{\\pi}4-\\theta)\\sin^2(\\dfrac{\\pi}4+\\theta)}{\\dfrac 12-\\cos^2\\theta}$.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76930,7 +76972,9 @@
     "003125": {
         "id": "003125",
         "content": "*求证: $\\dfrac{2\\cos\\alpha}{1+\\sin\\alpha+\\cos\\alpha}=1-\\tan\\dfrac{\\alpha}2$.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76951,7 +76995,9 @@
     "003126": {
         "id": "003126",
         "content": "化简: $\\sin^2\\alpha\\sin^2\\beta+\\cos^2\\alpha\\cos^2\\beta-\\dfrac 12\\cos2\\alpha\\cos 2\\beta$.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -76972,7 +77018,9 @@
     "003127": {
         "id": "003127",
         "content": "已知$0<\\alpha<\\dfrac{\\pi}4$, 且$\\dfrac{2\\sin^2\\alpha+\\sin 2\\alpha}{1+\\tan \\alpha}=k$, 分别用$k$表示$\\sin\\alpha\\cdot \\cos\\alpha$及$\\sin\\alpha-\\cos\\alpha$.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -109414,7 +109462,9 @@
     "004548": {
         "id": "004548",
         "content": "已知$A(\\cos \\alpha,\\sin\\alpha)$, $B(\\cos\\beta,\\sin\\beta)$, 其中$\\alpha,\\beta$为锐角, 且$|AB|=\\dfrac{\\sqrt{10}}5$.\\\\\n(1) 求$\\cos(\\alpha-\\beta)$的值;\\\\\n(2) 若$\\tan\\dfrac\\alpha 2=\\dfrac 12$, 求$\\cos\\alpha$及$\\cos\\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0309001B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -112734,7 +112784,9 @@
     "004681": {
         "id": "004681",
         "content": "若实数$x,y\\in [0,2\\pi]$, 且满足$\\cos (x+y)=\\cos x+\\cos y$, 则称$x$与$y$是``余弦相关''的.\\\\\n(1) 若$x=\\dfrac{\\pi}2$, 求出所有与之``余弦相关''的实数$y$;\\\\\n(2) 若存在实数$y$, 与$x$``余弦相关'', 求$x$的取值范围;\\\\\n(3) 若不相等的两个实数$x$与$y$是``余弦相关''的, 求证: 存在实数$z$, 使得$x$与$z$ 为``余弦相关''的, $y$与$z$也为``余弦相关''的.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145568,7 +145620,9 @@
     "006100": {
         "id": "006100",
         "content": "已知$\\cos (\\alpha +\\beta)=\\dfrac 45$, $\\cos (\\alpha -\\beta)=-\\dfrac 45$, 其中$\\alpha +\\beta \\in (\\dfrac{7\\pi}4,2\\pi)$, $\\alpha -\\beta \\in (\\dfrac{3\\pi}4,\\pi)$, 求$\\cos 2\\alpha$.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145589,7 +145643,9 @@
     "006101": {
         "id": "006101",
         "content": "求证: $\\tan (\\alpha -\\beta)+\\tan (\\beta -\\gamma)+\\tan (\\gamma -\\alpha)=\\tan (\\alpha -\\beta)\\tan (\\beta -\\gamma)\\tan (\\gamma -\\alpha)$.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145652,7 +145708,9 @@
     "006104": {
         "id": "006104",
         "content": "求$\\tan 65^\\circ +\\tan 70^\\circ +1-\\tan 65^\\circ \\tan 70^\\circ$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145887,7 +145945,9 @@
     "006115": {
         "id": "006115",
         "content": "若锐角$\\alpha ,\\beta$满足$\\cos \\alpha =\\dfrac 35$, $\\cos (\\alpha +\\beta)=-\\dfrac 5{13}$则$\\cos \\beta =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145908,7 +145968,9 @@
     "006116": {
         "id": "006116",
         "content": "若$\\cos (\\alpha -\\beta)=-\\dfrac 45$, $\\cos (\\alpha +\\beta)=\\dfrac 45$, 且$90^\\circ <\\alpha -\\beta <180^\\circ$, $270^\\circ <\\alpha +\\beta <360^\\circ$, 则$\\cos 2\\alpha =$\\blank{50}, $\\cos 2\\beta =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145929,7 +145991,9 @@
     "006117": {
         "id": "006117",
         "content": "若$\\cos x+\\cos y=\\dfrac 12$, $\\sin x-\\sin y=\\dfrac 13$, 则$\\cos (x+y)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -145992,7 +146056,9 @@
     "006120": {
         "id": "006120",
         "content": "若$\\sin \\alpha +\\sin \\beta =\\dfrac{\\sqrt 2}2$, 则$\\cos \\alpha +\\cos \\beta$的取值范围是\\bracket{20}.\n\\fourch{$[0,\\dfrac{\\sqrt 2}2]$}{$[-\\dfrac{\\sqrt 2}2,\\dfrac{\\sqrt 2}2]$}{$[-2, 2]$}{$[-\\dfrac{\\sqrt {14}}2,\\dfrac{\\sqrt {14}}2]$.}",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146055,7 +146121,9 @@
     "006123": {
         "id": "006123",
         "content": "已知锐角$\\alpha ,\\beta$满足$\\cos \\alpha =\\dfrac 45$, $\\tan (\\alpha -\\beta)=-\\dfrac 13$, 求$\\cos \\beta$.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146076,7 +146144,9 @@
     "006124": {
         "id": "006124",
         "content": "已知$\\cos (\\dfrac{\\pi}4-\\alpha)=\\dfrac 35$, $\\sin (\\dfrac{3\\pi}4+\\beta)=\\dfrac 5{13}$, 其中$\\dfrac{\\pi}4<\\alpha <\\dfrac{3\\pi}4$, $0<\\beta <\\dfrac{\\pi}4$, 求$\\sin (\\alpha +\\beta)$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146097,7 +146167,9 @@
     "006125": {
         "id": "006125",
         "content": "已知$\\alpha ,\\beta$为锐角, 满足$\\cos \\alpha =\\dfrac 17$, $\\sin (\\alpha +\\beta)=\\dfrac{5\\sqrt 3}{14}$, 求$\\cos \\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146118,7 +146190,9 @@
     "006126": {
         "id": "006126",
         "content": "已知$8\\cos (2\\alpha +\\beta)+5\\cos \\beta =0$, 求$\\tan (\\alpha +\\beta)\\cdot \\tan \\alpha$的值.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146160,7 +146234,9 @@
     "006128": {
         "id": "006128",
         "content": "已知锐角$\\alpha ,\\beta ,\\gamma$满足$\\sin \\alpha +\\sin \\gamma =\\sin \\beta$, $\\cos \\alpha -\\cos \\gamma =\\cos \\beta$, 求$\\alpha -\\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0309003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146181,7 +146257,9 @@
     "006129": {
         "id": "006129",
         "content": "若$\\alpha ,\\beta$为锐角, 且满足$\\cos \\alpha =\\dfrac 45$, $\\cos (\\alpha +\\beta)=\\dfrac 35$, 则$\\sin \\beta$的值是\\bracket{20}.\n\\fourch{$\\dfrac{17}{25}$}{$\\dfrac 35$}{$\\dfrac 7{25}$}{$\\dfrac 15$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146370,7 +146448,9 @@
     "006138": {
         "id": "006138",
         "content": "计算: $\\dfrac{\\sin 7^\\circ +\\sin 8^\\circ \\cos 15^\\circ}{\\cos 7^\\circ -\\sin 8^\\circ \\sin 15^\\circ}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146521,7 +146601,9 @@
     "006145": {
         "id": "006145",
         "content": "已知关于$x$的方程$x^2+px+q=0$的两根是$\\tan \\alpha$, $\\tan \\beta$, 求$\\dfrac{\\sin (\\alpha +\\beta)}{\\cos (\\alpha -\\beta)}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146542,7 +146624,9 @@
     "006146": {
         "id": "006146",
         "content": "已知$\\sin (\\alpha +\\beta)=\\dfrac 12$, $\\sin (\\alpha -\\beta)=\\dfrac 13$, 求$\\tan \\alpha \\cot \\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146566,7 +146650,9 @@
     "006147": {
         "id": "006147",
         "content": "已知$\\tan (\\alpha +\\beta)=-2$, $\\tan (\\alpha -\\beta)=\\dfrac 12$, 求$\\dfrac{\\sin 2\\alpha}{\\sin 2\\beta}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146587,7 +146673,9 @@
     "006148": {
         "id": "006148",
         "content": "已知$\\tan \\alpha =1$, $3\\sin \\beta =\\sin (2\\alpha +\\beta)$, 求$\\tan (\\alpha +\\beta)$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146608,7 +146696,9 @@
     "006149": {
         "id": "006149",
         "content": "已知$\\dfrac{\\tan (\\alpha -\\gamma)}{\\tan \\alpha}+\\dfrac{\\sin ^2\\beta}{\\sin ^2\\alpha}=1$, 求证: $\\tan ^2\\beta =\\tan \\alpha \\tan \\gamma$.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146673,7 +146763,9 @@
     "006152": {
         "id": "006152",
         "content": "若$\\tan (\\alpha +\\beta)=\\dfrac 25$, $\\tan (\\beta -\\dfrac{\\pi}4)=\\dfrac 14$, 则$\\tan (\\alpha +\\dfrac{\\pi}4)$等于\\bracket{20}.\n\\fourch{$\\dfrac{13}{18}$}{$\\dfrac{13}{22}$}{$\\dfrac 3{22}$}{$\\dfrac 16$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146694,7 +146786,9 @@
     "006153": {
         "id": "006153",
         "content": "若$\\dfrac{1-\\tan A}{1+\\tan A}=4+\\sqrt 5$, 则$\\cot (\\dfrac{\\pi}4+A)$的值等于\\bracket{20}.\n\\fourch{$-4-\\sqrt 5$}{$4+\\sqrt 5$}{$-\\dfrac 1{4+\\sqrt 5}$}{$\\dfrac 1{4+\\sqrt 5}$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146715,7 +146809,9 @@
     "006154": {
         "id": "006154",
         "content": "已知$\\alpha +\\beta =\\dfrac{3\\pi}4$, 则$(1-\\tan \\alpha)(1-\\tan \\beta)$的值等于\\bracket{20}.\n\\fourch{$2$}{$-2$}{$1$}{$-1$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146736,7 +146832,9 @@
     "006155": {
         "id": "006155",
         "content": "计算$\\dfrac{1+\\cot 15^\\circ}{1-\\tan 75^\\circ}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146757,7 +146855,9 @@
     "006156": {
         "id": "006156",
         "content": "若$\\alpha +\\beta =\\dfrac{\\pi}4$, 则$\\dfrac{1-\\tan \\beta}{1+\\tan \\beta}=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146778,7 +146878,9 @@
     "006157": {
         "id": "006157",
         "content": "若$\\tan x=\\dfrac 12$, $\\tan (x-y)=-\\dfrac 25$, 则$\\tan (2x-y)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146799,7 +146901,9 @@
     "006158": {
         "id": "006158",
         "content": "在$\\triangle ABC$中, $\\tan A$, $\\tan B$是方程$3x^2+8x-1=0$的两个根, 则$\\tan C=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146820,7 +146924,9 @@
     "006159": {
         "id": "006159",
         "content": "若$\\tan (\\alpha +\\dfrac{\\pi}4)=-\\dfrac 9{40}$, 则$\\tan \\alpha =$\\blank{50}, $\\tan (\\alpha -\\dfrac{\\pi}4)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146883,7 +146989,9 @@
     "006162": {
         "id": "006162",
         "content": "若$-\\dfrac{\\pi}2<\\alpha <\\dfrac{\\pi}2$, $-\\dfrac{\\pi}2<\\beta <\\dfrac{\\pi}2$, 且$\\tan \\alpha$, $\\tan \\beta$是方程$x^2+3\\sqrt 3x+4=0$的两个根, 则$\\alpha +\\beta$等于\\bracket{20}.\n\\fourch{$\\dfrac{\\pi}3$}{$-\\dfrac{2\\pi}3$}{$\\dfrac{\\pi}3$或$\\dfrac{4\\pi}3$}{$\\dfrac{\\pi}3$或$-\\dfrac{2\\pi}3$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146904,7 +147012,9 @@
     "006163": {
         "id": "006163",
         "content": "若$\\tan \\theta$和$\\tan (\\dfrac{\\pi}4-\\theta)$是方程$x^2+px+q=0$的两个根, 则$p,q$满足关系式\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146925,7 +147035,9 @@
     "006164": {
         "id": "006164",
         "content": "若$\\tan \\alpha =\\dfrac 17$, $\\tan \\beta =\\dfrac 13$, $\\alpha ,\\beta \\in (0,\\dfrac{\\pi}2)$, 则$\\alpha +2\\beta =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146946,7 +147058,9 @@
     "006165": {
         "id": "006165",
         "content": "计算: $1+\\tan 66^\\circ +\\tan 69^\\circ -\\tan 66^\\circ \\tan 69^\\circ =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146967,7 +147081,9 @@
     "006166": {
         "id": "006166",
         "content": "计算: $\\tan 19^\\circ +\\tan 101^\\circ -\\sqrt 3\\tan 19^\\circ \\tan 101^\\circ =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -146988,7 +147104,9 @@
     "006167": {
         "id": "006167",
         "content": "若$\\alpha +\\beta =k\\pi +\\dfrac{\\pi}4$($k\\in \\mathbf{Z}$), 则$(1+\\tan \\alpha)(1+\\tan \\beta)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147009,7 +147127,9 @@
     "006168": {
         "id": "006168",
         "content": "计算$(1+\\tan 1^\\circ)(1+\\tan 2^\\circ)(1+\\tan 3^\\circ)\\cdots (1+\\tan 43^\\circ)(1+\\tan 44^\\circ)=$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147030,7 +147150,9 @@
     "006169": {
         "id": "006169",
         "content": "求证: $\\tan 20^\\circ \\tan 30^\\circ +\\tan 30^\\circ \\tan 40^\\circ +\\tan 40^\\circ \\tan 20^\\circ =1$.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147051,7 +147173,9 @@
     "006170": {
         "id": "006170",
         "content": "求证: 当$A+B+C=k\\pi$($k\\in \\mathbf{Z}$)时, $\\tan (A-B)+\\tan (B-C)+\\tan (C-A)=\\tan (A-B)\\tan (B-C)\\tan (C-A)$.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147072,7 +147196,9 @@
     "006171": {
         "id": "006171",
         "content": "求证: $\\tan A+\\tan B+\\tan C=\\tan A\\tan B\\tan C$, 其中$A+B+C=k\\pi$($k\\in \\mathbf{Z}$).",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147093,7 +147219,9 @@
     "006172": {
         "id": "006172",
         "content": "已知锐角$\\alpha ,\\beta$满足$\\tan \\alpha =\\sqrt 3(m+1)$, $\\tan (-\\beta)=\\sqrt 3(\\tan \\alpha \\tan \\beta +m)$, 求$\\alpha +\\beta$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147114,7 +147242,9 @@
     "006173": {
         "id": "006173",
         "content": "求$\\dfrac{\\tan 20^\\circ +\\tan 40^\\circ +\\tan 120^\\circ}{\\tan 20^\\circ \\tan 40^\\circ}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147135,7 +147265,9 @@
     "006174": {
         "id": "006174",
         "content": "已知$\\tan \\theta =\\dfrac{\\sin \\alpha -\\cos \\alpha}{\\sin \\alpha +\\cos \\alpha}$($\\alpha ,\\theta$都是锐角), 求$\\dfrac{\\sin \\alpha -\\cos \\alpha}{\\sin \\theta}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147156,7 +147288,9 @@
     "006175": {
         "id": "006175",
         "content": "已知$\\tan (\\dfrac{\\pi}4+\\alpha)=-\\dfrac 12$, 求$\\dfrac{2\\cos \\alpha (\\sin \\alpha -\\cos \\alpha)}{1+\\tan \\alpha}$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147177,7 +147311,9 @@
     "006176": {
         "id": "006176",
         "content": "已知$\\tan \\alpha$, $\\tan \\beta$是关于$x$的方程$mx^2-2x\\sqrt {7m-3}+2m=0$的两个实根, 求$\\tan (\\alpha +\\beta)$的取值范围.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147198,7 +147334,9 @@
     "006177": {
         "id": "006177",
         "content": "若$\\sin \\alpha +\\cos \\alpha =-\\sqrt 2$, 则$\\tan \\alpha +\\cot \\alpha$等于\\bracket{20}.\n\\fourch{$-2$}{$-1$}{$1$}{$2$}",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147471,7 +147609,9 @@
     "006190": {
         "id": "006190",
         "content": "若$\\sin 2\\alpha =\\dfrac 45$, 则$\\tan ^2\\alpha +\\cot ^2\\alpha =$\\blank{50}.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147494,7 +147634,9 @@
     "006191": {
         "id": "006191",
         "content": "若$\\sin x\\cos y=\\dfrac 12$, 则$\\cos x\\sin y$的取值范围是\\bracket{20}.\n\\fourch{$[-\\dfrac 12,\\dfrac 12]$}{$[-\\dfrac 32,\\dfrac 12]$}{$[-\\dfrac 12,\\dfrac 32]$}{$[-1,1]$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -147979,7 +148121,9 @@
     "006214": {
         "id": "006214",
         "content": "已知$\\sin \\alpha +\\sin \\beta =\\dfrac 12$, $\\cos \\alpha +\\cos \\beta =\\dfrac 13$, 求$\\cos ^2(\\dfrac{\\alpha -\\beta}2)$的值.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -148368,7 +148512,9 @@
     "006232": {
         "id": "006232",
         "content": "已知$\\sin \\alpha =\\dfrac 35$, $\\alpha \\in (\\dfrac{\\pi}2,\\pi)$, $\\tan (\\pi -\\beta)=\\dfrac 12$, 求$\\tan (\\alpha -2\\beta)$的值.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -148895,7 +149041,9 @@
     "006257": {
         "id": "006257",
         "content": "若$\\cos ^2\\alpha -\\cos ^2\\beta =m$, 则$\\sin (\\alpha +\\beta)\\sin (\\alpha -\\beta)$等于\\bracket{20}.\n\\fourch{$4m$}{$-4m$}{$m$}{$-m$}",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -149509,7 +149657,9 @@
     "006286": {
         "id": "006286",
         "content": "下列各式中, 不正确的是\\bracket{20}.\n\\twoch{$\\sin \\alpha +\\sin \\beta =2\\sin \\dfrac{\\beta +\\alpha}2\\cos \\dfrac{\\beta -\\alpha}2$}{$\\sin \\alpha -\\sin \\beta =2\\cos \\dfrac{\\beta +\\alpha}2\\sin \\dfrac{\\beta -\\alpha}2$}{$\\cos \\alpha +\\cos \\beta =2\\cos \\dfrac{\\beta +\\alpha}2\\cos \\dfrac{\\beta -\\alpha}2$}{$\\cos \\alpha -\\cos \\beta =2\\sin \\dfrac{\\beta +\\alpha}2\\sin \\dfrac{\\beta -\\alpha}2$}",
-        "objs": [],
+        "objs": [
+            "K0313003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -149551,7 +149701,9 @@
     "006288": {
         "id": "006288",
         "content": "将$\\cos ^2x-\\sin ^2y$化为积的形式, 结果是\\bracket{20}.\n\\fourch{$-\\sin (x+y)\\sin (x-y)$}{$\\cos (x+y)\\cos (x-y)$}{$\\sin (x+y)\\cos (x-y)$}{$-\\cos (x+y)\\sin (x-y)$}",
-        "objs": [],
+        "objs": [
+            "K0313003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -190609,7 +190761,9 @@
     "008168": {
         "id": "008168",
         "content": "证明: $\\sin (\\alpha +\\beta)\\cos (\\alpha -\\beta)=\\sin \\alpha \\cos \\alpha +\\sin \\beta \\cos \\beta$.",
-        "objs": [],
+        "objs": [
+            "K0313002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -190630,7 +190784,9 @@
     "008169": {
         "id": "008169",
         "content": "把$\\sqrt 3\\sin \\alpha +\\cos \\alpha$化成$A\\sin (\\alpha +\\varphi)$($A>0$)的形式.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -190651,7 +190807,9 @@
     "008170": {
         "id": "008170",
         "content": "把$5\\sin \\alpha -12\\cos \\alpha$化成$A\\sin (\\alpha +\\varphi)$($A>0$)的形式.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -190791,7 +190949,9 @@
     "008176": {
         "id": "008176",
         "content": "已知$\\tan \\alpha$、$\\tan \\beta$是方程$x^2+6x+7=0$的两个根, 求证: $\\sin (\\alpha +\\beta)=\\cos (\\alpha +\\beta)$.",
-        "objs": [],
+        "objs": [
+            "K0310002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -195391,7 +195551,9 @@
     "008385": {
         "id": "008385",
         "content": "证明: $(\\sin \\alpha +\\sin \\beta)^2+(\\cos \\alpha +\\cos \\beta)^2=4\\cos ^2\\dfrac{\\alpha -\\beta}2$.",
-        "objs": [],
+        "objs": [
+            "K0312003B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -221077,7 +221239,9 @@
     "009572": {
         "id": "009572",
         "content": "把下列各式化为$A\\sin (\\alpha+\\varphi )$($A>0$)的形式:\\\\\n(1) $\\sin \\alpha+\\cos \\alpha$;\\\\\n(2) $-\\sin \\alpha+\\sqrt 3\\cos \\alpha$.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],
@@ -236002,7 +236166,9 @@
     "010251": {
         "id": "010251",
         "content": "把下列各式化成$A\\sin (\\alpha+\\varphi)$($A>0$)的形式:\\\\\n(1) $\\sqrt 3\\sin \\alpha+\\cos \\alpha$;\\\\\n(2) $5\\sin \\alpha-12\\cos \\alpha$.",
-        "objs": [],
+        "objs": [
+            "K0311002B"
+        ],
         "tags": [
             "第三单元"
         ],