From e220fc9af17c8a461b7cc0e77907d549cef9a8bd Mon Sep 17 00:00:00 2001
From: wangweiye7840 <kj_wangweiye@126.com>
Date: Tue, 19 Mar 2024 14:04:52 +0800
Subject: [PATCH] =?UTF-8?q?K03=E5=8D=95=E5=85=83=E7=9A=84=E8=AF=BE?=
 =?UTF-8?q?=E6=97=B6=E7=9B=AE=E6=A0=87=E6=B7=BB=E5=8A=A0predecessor?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
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---
 题库0.3/LessonObj.json | 350 ++++++++++++++++++++++++++++++++---------
 1 file changed, 275 insertions(+), 75 deletions(-)

diff --git a/题库0.3/LessonObj.json b/题库0.3/LessonObj.json
index c64a4602..c6efdacc 100644
--- a/题库0.3/LessonObj.json
+++ b/题库0.3/LessonObj.json
@@ -1555,241 +1555,349 @@
         "id": "K0301002B",
         "unit_obj": "D03001B",
         "content": "会判断角在平面直角坐标系中的位置.",
-        "predecessor": []
+        "predecessor": [
+            "K0301001B"
+        ]
     },
     "K0301003B": {
         "id": "K0301003B",
         "unit_obj": "D03001B",
         "content": "角的加减运算与角终边的旋转之间的关系.",
-        "predecessor": []
+        "predecessor": [
+            "K0301001B",
+            "K0301002B"
+        ]
     },
     "K0301004B": {
         "id": "K0301004B",
         "unit_obj": "D03001B",
         "content": "终边有特殊位置关系的角之间的等量关系.",
-        "predecessor": []
+        "predecessor": [
+            "K0301003B"
+        ]
     },
     "K0302001B": {
         "id": "K0302001B",
         "unit_obj": "D03001B",
         "content": "了解弧度制, 能进行一般的角度制与弧度制的转化.",
-        "predecessor": []
+        "predecessor": [
+            "K0301001B"
+        ]
     },
     "K0302002B": {
         "id": "K0302002B",
         "unit_obj": "D03001B",
         "content": "掌握弧度制下扇形的弧长和面积公式.",
-        "predecessor": []
+        "predecessor": [
+            "K0302001B"
+        ]
     },
     "K0302003B": {
         "id": "K0302003B",
         "unit_obj": "D03001B",
         "content": "在弧度制下会用代数方法表示和研究角.",
-        "predecessor": []
+        "predecessor": [
+            "K0302001B"
+        ]
     },
     "K0303001B": {
         "id": "K0303001B",
         "unit_obj": "D03001B",
         "content": "掌握任意角的用比值给出的正弦、余弦、正切、余切的定义.",
-        "predecessor": []
+        "predecessor": [
+            "K0301001B"
+        ]
     },
     "K0303002B": {
         "id": "K0303002B",
         "unit_obj": "D03001B",
         "content": "掌握不同象限的角的正弦、余弦、正切和余切的符号.",
-        "predecessor": []
+        "predecessor": [
+            "K0301002B",
+            "K0303001B"
+        ]
     },
     "K0304001B": {
         "id": "K0304001B",
         "unit_obj": "D03001B",
         "content": "理解角的终边和单位圆的交点的坐标与角的正弦、余弦、正切和余切的关系.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B"
+        ]
     },
     "K0304002B": {
         "id": "K0304002B",
         "unit_obj": "D03001B",
         "content": "经历$\\sin^2\\alpha+\\cos^2\\alpha=1$; $\\tan\\alpha=\\dfrac{\\sin\\alpha}{\\cos\\alpha}$; $\\cot\\alpha=\\dfrac{\\cos\\alpha}{\\sin\\alpha}$; $\\tan\\alpha\\cdot \\cot\\alpha=1$的推导.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B"
+        ]
     },
     "K0304003B": {
         "id": "K0304003B",
         "unit_obj": "D03001B",
         "content": "会用$\\sin^2\\alpha+\\cos^2\\alpha=1$; $\\tan\\alpha=\\dfrac{\\sin\\alpha}{\\cos\\alpha}$; $\\cot\\alpha=\\dfrac{\\cos\\alpha}{\\sin\\alpha}$; $\\tan\\alpha\\cdot \\cot\\alpha=1$解决``已知一个三角比的值, 求其他三角比的值''的问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0304002B",
+            "K0303002B"
+        ]
     },
     "K0305001B": {
         "id": "K0305001B",
         "unit_obj": "D03001B",
         "content": "会用同角三角函数的基本关系式($\\sin^2\\alpha+\\cos^2\\alpha=1$; $\\tan\\alpha=\\dfrac{\\sin\\alpha}{\\cos\\alpha}$; $\\cot\\alpha=\\dfrac{\\cos\\alpha}{\\sin\\alpha}$; $\\tan\\alpha\\cdot \\cot\\alpha=1$), 在熟悉的情境中, 解决一些三角恒等式的化简与证明问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0304002B",
+            "K0303002B"
+        ]
     },
     "K0306001B": {
         "id": "K0306001B",
         "unit_obj": "D03002B",
         "content": "借助单位圆的对称性, 经历利用定义推导出第一组诱导公式(有关$k\\pi\\pm \\alpha$)的正弦、余弦、正切和余切的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0301004B",
+            "K0304001B"
+        ]
     },
     "K0306002B": {
         "id": "K0306002B",
         "unit_obj": "D03002B",
         "content": "会利用第一组诱导公式(有关$k\\pi\\pm \\alpha$)进行简单的求值、化简与证明.",
-        "predecessor": []
+        "predecessor": [
+            "K0306001B"
+        ]
     },
     "K0307001B": {
         "id": "K0307001B",
         "unit_obj": "D03002B",
         "content": "借助单位圆的对称性, 经历利用定义推导出第二组诱导公式(有关$(k+\\dfrac 12)\\pi\\pm \\alpha$)的正弦、余弦、正切和余切的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0301004B",
+            "K0304001B"
+        ]
     },
     "K0307002B": {
         "id": "K0307002B",
         "unit_obj": "D03002B",
         "content": "会通过``奇变偶不变, 符号看象限''来记忆诱导公式.",
-        "predecessor": []
+        "predecessor": [
+            "K0306001B",
+            "K0307001B"
+        ]
     },
     "K0307003B": {
         "id": "K0307003B",
         "unit_obj": "D03002B",
         "content": "会利用第二组诱导公式(有关$(k+\\dfrac 12)\\pi\\pm \\alpha$)进行简单的求值、化简与证明.",
-        "predecessor": []
+        "predecessor": [
+            "K0307001B"
+        ]
     },
     "K0307004B": {
         "id": "K0307004B",
         "unit_obj": "D03002B",
         "content": "理解可以通过终边的旋转、对称等方式, 利用诱导公式研究平面上的坐标变换.",
-        "predecessor": []
+        "predecessor": [
+            "K0301004B",
+            "K0306001B",
+            "K0307001B",
+            "K0304001B"
+        ]
     },
     "K0308001B": {
         "id": "K0308001B",
         "unit_obj": "D03002B",
         "content": "能够从已知特殊三角值的角的正弦、余弦、正切值求角的集合, 并能简单应用.",
-        "predecessor": []
+        "predecessor": [
+            "K0306001B",
+            "K0307001B",
+            "K0304001B"
+        ]
     },
     "K0308002B": {
         "id": "K0308002B",
         "unit_obj": "D03002B",
         "content": "能借助角的三角比的特殊值解简单的三角方程.",
-        "predecessor": []
+        "predecessor": [
+            "K0308001B"
+        ]
     },
     "K0308003B": {
         "id": "K0308003B",
         "unit_obj": "D03002B",
         "content": "掌握锐角的反三角函数表示, 并能用计算器求出近似值.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B"
+        ]
     },
     "K0308004B": {
         "id": "K0308004B",
         "unit_obj": "D03002B",
         "content": "借助单位圆, 能用反三角符号表示的锐角表示一般角.",
-        "predecessor": []
+        "predecessor": [
+            "K0308003B",
+            "K0304001B"
+        ]
     },
     "K0308005B": {
         "id": "K0308005B",
         "unit_obj": "D03002B",
         "content": "会解具体的最简三角方程(指$\\sin x=\\sin \\alpha$, $\\cos x=\\cos \\alpha$, $\\tan x=\\tan\\alpha$).",
-        "predecessor": []
+        "predecessor": [
+            "K0308004B",
+            "K0308002B"
+        ]
     },
     "K0309001B": {
         "id": "K0309001B",
         "unit_obj": "D03002B",
         "content": "经历两角差的余弦公式的坐标法的推导过程, 知道两角差的余弦公式的意义.",
-        "predecessor": []
+        "predecessor": [
+            "K0304001B"
+        ]
     },
     "K0309002B": {
         "id": "K0309002B",
         "unit_obj": "D03002B",
         "content": "两角差的余弦推导两角和的余弦.",
-        "predecessor": []
+        "predecessor": [
+            "K0309001B",
+            "K0306002B"
+        ]
     },
     "K0309003B": {
         "id": "K0309003B",
         "unit_obj": "D03002B",
         "content": "会灵活选择角, 用两角和差的余弦公式求值及化简.",
-        "predecessor": []
+        "predecessor": [
+            "K0309001B",
+            "K0309002B"
+        ]
     },
     "K0310001B": {
         "id": "K0310001B",
         "unit_obj": "D03002B",
         "content": "了解两角差的余弦公式推导两角和与差的正弦、正切公式的路径, 并经历推导过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0309001B",
+            "K0307003B",
+            "K0306002B"
+        ]
     },
     "K0310002B": {
         "id": "K0310002B",
         "unit_obj": "D03002B",
         "content": "会灵活选择角, 用两角和、差的正弦和正切公式求值及化简.",
-        "predecessor": []
+        "predecessor": [
+            "K0310001B"
+        ]
     },
     "K0311001B": {
         "id": "K0311001B",
         "unit_obj": "D03002B",
         "content": "理解可以通过终边的旋转、对称等方式, 利用两角和、差的正弦、余弦公式研究平面上的坐标变换.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B",
+            "K0301004B",
+            "K0310002B",
+            "K0309003B"
+        ]
     },
     "K0311002B": {
         "id": "K0311002B",
         "unit_obj": "D03002B",
         "content": "会用辅助角公式将$a\\sin\\alpha+b\\cos\\alpha$型的表达式整理为$A\\sin(\\alpha+\\varphi)$及$A\\cos(\\alpha+\\varphi)$的形式, 并能明确给出辅助角$\\varphi$的正弦与余弦值.",
-        "predecessor": []
+        "predecessor": [
+            "K0310002B",
+            "K0309003B"
+        ]
     },
     "K0312001B": {
         "id": "K0312001B",
         "unit_obj": "D03002B",
         "content": "经历利用两角和公式, 推导出二倍角的正弦、余弦、正切公式的过程, 并了解它们的内在联系.",
-        "predecessor": []
+        "predecessor": [
+            "K0310002B",
+            "K0309003B"
+        ]
     },
     "K0312002B": {
         "id": "K0312002B",
         "unit_obj": "D03002B",
         "content": "熟悉二倍角余弦公式的三种不同形式.",
-        "predecessor": []
+        "predecessor": [
+            "K0312001B",
+            "K0305001B"
+        ]
     },
     "K0312003B": {
         "id": "K0312003B",
         "unit_obj": "D03002B",
         "content": "利用二倍角公式, 进行求值、化简和证明.",
-        "predecessor": []
+        "predecessor": [
+            "K0312002B"
+        ]
     },
     "K0313001B": {
         "id": "K0313001B",
         "unit_obj": "D03002B",
         "content": "能运用所学公式进行简单的恒等变换, 推导半角公式.",
-        "predecessor": []
+        "predecessor": [
+            "K0312002B"
+        ]
     },
     "K0313002B": {
         "id": "K0313002B",
         "unit_obj": "D03002B",
         "content": "能运用所学公式进行简单的恒等变换, 推导积化和差公式.",
-        "predecessor": []
+        "predecessor": [
+            "K0309003B",
+            "K0310002B"
+        ]
     },
     "K0313003B": {
         "id": "K0313003B",
         "unit_obj": "D03002B",
         "content": "能运用所学公式进行简单的恒等变换, 推导和差化积公式.",
-        "predecessor": []
+        "predecessor": [
+            "K0309003B",
+            "K0310002B",
+            "K0313002B"
+        ]
     },
     "K0314001B": {
         "id": "K0314001B",
         "unit_obj": "D03003B",
         "content": "经历用坐标法推导三角形面积公式$S=\\dfrac{1}{2}ab\\sin C$等的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B"
+        ]
     },
     "K0314002B": {
         "id": "K0314002B",
         "unit_obj": "D03003B",
         "content": "利用三角形面积公式推导得到正弦定理.",
-        "predecessor": []
+        "predecessor": [
+            "K0314001B"
+        ]
     },
     "K0314003B": {
         "id": "K0314003B",
         "unit_obj": "D03003B",
         "content": "会用正弦定理解决``ASA''型的解三角形问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0314002B"
+        ]
     },
     "K0314004B": {
         "id": "K0314004B",
         "unit_obj": "D03003B",
         "content": "会用正弦定理及面积公式证明三角形中关于边、角和面积的恒等式.",
-        "predecessor": []
+        "predecessor": [
+            "K0314001B",
+            "K0314002B"
+        ]
     },
     "K0314005B": {
         "id": "K0314005B",
@@ -1801,163 +1909,234 @@
         "id": "K0314006B",
         "unit_obj": "D03003B",
         "content": "利用圆周角均相等推导含$2R$的正弦定理的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0314005B"
+        ]
     },
     "K0315001B": {
         "id": "K0315001B",
         "unit_obj": "D03003B",
         "content": "经历用坐标法推导余弦定理的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0314001B"
+        ]
     },
     "K0315002B": {
         "id": "K0315002B",
         "unit_obj": "D03003B",
         "content": "熟悉并记忆余弦定理.",
-        "predecessor": []
+        "predecessor": [
+            "K0315001B"
+        ]
     },
     "K0315003B": {
         "id": "K0315003B",
         "unit_obj": "D03003B",
         "content": "会用余弦定理解``SSS''``SAS''型的解三角形问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0315002B"
+        ]
     },
     "K0315004B": {
         "id": "K0315004B",
         "unit_obj": "D03003B",
         "content": "能够灵活运用正弦定理、余弦定理解决``SSA''型的解三角形问题, 并能正确取舍解得结果.",
-        "predecessor": []
+        "predecessor": [
+            "K0314003B",
+            "K0315003B"
+        ]
     },
     "K0316001B": {
         "id": "K0316001B",
         "unit_obj": "D03003B",
         "content": "会灵活运用正弦定理和余弦定理证明三角形中的等式.",
-        "predecessor": []
+        "predecessor": [
+            "K0314002B",
+            "K0315002B"
+        ]
     },
     "K0316002B": {
         "id": "K0316002B",
         "unit_obj": "D03003B",
         "content": "会灵活运用正弦定理和余弦定理判断三角形的形状.",
-        "predecessor": []
+        "predecessor": [
+            "K0314003B",
+            "K0315003B",
+            "K0315004B"
+        ]
     },
     "K0317001B": {
         "id": "K0317001B",
         "unit_obj": "D03003B",
         "content": "能用正弦定理、余弦定理解决简单的实际问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0314003B",
+            "K0315003B",
+            "K0315004B"
+        ]
     },
     "K0317002B": {
         "id": "K0317002B",
         "unit_obj": "D03003B",
         "content": "能将有关测量的问题转化为解三角形问题, 并灵活运用正弦定理和余弦定理求解.",
-        "predecessor": []
+        "predecessor": [
+            "K0317001B"
+        ]
     },
     "K0318001B": {
         "id": "K0318001B",
         "unit_obj": "D03004B",
         "content": "建立正弦函数的概念.",
-        "predecessor": []
+        "predecessor": [
+            "K0303001B",
+            "K0215001B"
+        ]
     },
     "K0318002B": {
         "id": "K0318002B",
         "unit_obj": "D03004B",
         "content": "经历描点, 平移绘制正弦函数图像的过程.",
-        "predecessor": []
+        "predecessor": [
+            "K0318001B",
+            "K0216003B"
+        ]
     },
     "K0318003B": {
         "id": "K0318003B",
         "unit_obj": "D03004B",
         "content": "会用五点法绘制正弦函数、与正弦函数相关的函数的大致图像.",
-        "predecessor": []
+        "predecessor": [
+            "K0318002B"
+        ]
     },
     "K0319001B": {
         "id": "K0319001B",
         "unit_obj": "D03004B",
         "content": "结合诱导公式理解正弦函数的周期性.",
-        "predecessor": []
+        "predecessor": [
+            "K0306002B"
+        ]
     },
     "K0319002B": {
         "id": "K0319002B",
         "unit_obj": "D03004B",
         "content": "直观地理解周期函数的定义.",
-        "predecessor": []
+        "predecessor": [
+            "K0319001B",
+            "K0318002B"
+        ]
     },
     "K0319003B": {
         "id": "K0319003B",
         "unit_obj": "D03004B",
         "content": "能用``数学语言''准确地给出周期函数的定义.",
-        "predecessor": []
+        "predecessor": [
+            "K0319001B"
+        ]
     },
     "K0319004B": {
         "id": "K0319004B",
         "unit_obj": "D03004B",
         "content": "会证明正弦函数的最小正周期是$2\\pi$.",
-        "predecessor": []
+        "predecessor": [
+            "K0319003B",
+            "K0318002B"
+        ]
     },
     "K0319005B": {
         "id": "K0319005B",
         "unit_obj": "D03004B",
         "content": "了解函数$y=A\\sin(\\omega x+\\varphi)$的周期, 并会粗略地说明道理.",
-        "predecessor": []
+        "predecessor": [
+            "K0319003B",
+            "K0306002B"
+        ]
     },
     "K0320001B": {
         "id": "K0320001B",
         "unit_obj": "D03004B",
         "content": "借助单位圆理解正弦函数的值域与最值.",
-        "predecessor": []
+        "predecessor": [
+            "K0304001B"
+        ]
     },
     "K0320002B": {
         "id": "K0320002B",
         "unit_obj": "D03004B",
         "content": "能运用正弦函数的值域与最值解决简单的正弦型函数的相应问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0320001B"
+        ]
     },
     "K0320003B": {
         "id": "K0320003B",
         "unit_obj": "D03004B",
         "content": "会将与正弦有关的现实情境中的问题转化为正弦函数的最值问题, 并加以解决.",
-        "predecessor": []
+        "predecessor": [
+            "K0320002B"
+        ]
     },
     "K0321001B": {
         "id": "K0321001B",
         "unit_obj": "D03004B",
         "content": "会判断并证明与正弦函数相关的函数的奇偶性.",
-        "predecessor": []
+        "predecessor": [
+            "K0306002B"
+        ]
     },
     "K0321002B": {
         "id": "K0321002B",
         "unit_obj": "D03004B",
         "content": "会借助单位圆及函数图像, 直观地理解正弦函数的单调性.",
-        "predecessor": []
+        "predecessor": [
+            "K0304001B",
+            "K0318002B"
+        ]
     },
     "K0321003B": {
         "id": "K0321003B",
         "unit_obj": "D03004B",
         "content": "能求$y=A\\sin(\\omega x+\\varphi)$型函数的单调区间, 其中$A>0$, $\\omega>0$.",
-        "predecessor": []
+        "predecessor": [
+            "K0321002B"
+        ]
     },
     "K0321004B": {
         "id": "K0321004B",
         "unit_obj": "D03004B",
         "content": "能求$y=A\\sin(\\omega x+\\varphi)$型函数的单调区间, 其中$A$与$\\omega$不全大于零.",
-        "predecessor": []
+        "predecessor": [
+            "K0321003B",
+            "K0307002B"
+        ]
     },
     "K0322001B": {
         "id": "K0322001B",
         "unit_obj": "D03004B",
         "content": "建立余弦函数的概念.",
-        "predecessor": []
+        "predecessor": [
+            "K0307003B"
+        ]
     },
     "K0322002B": {
         "id": "K0322002B",
         "unit_obj": "D03004B",
         "content": "借助正弦函数的相关性质, 掌握余弦函数的奇偶性、周期性、单调性、值域与最值等性质及其图像特征.",
-        "predecessor": []
+        "predecessor": [
+            "K0322001B",
+            "K0321001B",
+            "K0319004B",
+            "K0321002B",
+            "K0320001B"
+        ]
     },
     "K0322003B": {
         "id": "K0322003B",
         "unit_obj": "D03004B",
         "content": "会将与余弦函数有关的问题借助第二诱导公式转化为正弦函数有关的问题.",
-        "predecessor": []
+        "predecessor": [
+            "K0307003B"
+        ]
     },
     "K0323001B": {
         "id": "K0323001B",
@@ -1969,49 +2148,70 @@
         "id": "K0323002B",
         "unit_obj": "D03004B",
         "content": "会用三角函数解决简单的与周期变化有关的实际问题, 体会可利用三角函数构建刻画周期变化事物的数学模型.",
-        "predecessor": []
+        "predecessor": [
+            "K0323001B"
+        ]
     },
     "K0323003B": {
         "id": "K0323003B",
         "unit_obj": "D03004B",
         "content": "了解函数$y=A\\sin(\\omega x+\\varphi)$参数的变化对函数图像的影响.会用五点作图法作出函数的大致图像.",
-        "predecessor": []
+        "predecessor": [
+            "K0323001B",
+            "K0318003B"
+        ]
     },
     "K0324001B": {
         "id": "K0324001B",
         "unit_obj": "D03004B",
         "content": "类比正弦函数, 建立正切函数的概念.",
-        "predecessor": []
+        "predecessor": [
+            "K0318001B",
+            "K0303001B"
+        ]
     },
     "K0324002B": {
         "id": "K0324002B",
         "unit_obj": "D03004B",
         "content": "类比正弦函数, 借助单位圆画出正切函数的图像.",
-        "predecessor": []
+        "predecessor": [
+            "K0318002B",
+            "K0304001B"
+        ]
     },
     "K0324003B": {
         "id": "K0324003B",
         "unit_obj": "D03004B",
         "content": "直观地掌握正切函数的图像特征.",
-        "predecessor": []
+        "predecessor": [
+            "K0324002B"
+        ]
     },
     "K0324004B": {
         "id": "K0324004B",
         "unit_obj": "D03004B",
         "content": "直观地理解正切函数的周期性与值域.",
-        "predecessor": []
+        "predecessor": [
+            "K0306002B",
+            "K0324003B"
+        ]
     },
     "K0324005B": {
         "id": "K0324005B",
         "unit_obj": "D03004B",
         "content": "会用代数语言表示正切函数的奇偶性及单调性.",
-        "predecessor": []
+        "predecessor": [
+            "K0306002B",
+            "K0220002B"
+        ]
     },
     "K0324006B": {
         "id": "K0324006B",
         "unit_obj": "D03004B",
         "content": "能借助正切函数的单调性求$y=A\\tan(\\omega x+\\varphi)$的单调区间.",
-        "predecessor": []
+        "predecessor": [
+            "K0324005B"
+        ]
     },
     "K0401001X": {
         "id": "K0401001X",