Bridging British Education Virtual Academy

1. Course Basic Information 1. 课程基本信息

Course Name: 0112 math Henry 课程名称： 0112 数学亨利

Topic: Trigonometric Equations and Proving Identities 主题： 三角函数方程求解与恒等式证明

Date: N/A 日期： 未明确说明

Student: Henry 学生： Henry

Teaching Focus 教学重点

Reviewing solutions for tan equations, introducing the need to adjust solution ranges for equations like sin(2x), and applying Pythagorean identities to prove trigonometric identities.

复习tan方程的解法，引入求解sin(2x)等方程时调整解域范围的需求，以及应用毕达哥拉斯恒等式证明三角恒等式。

Teaching Objectives 教学目标

Successfully solve trigonometric equations, paying close attention to the periodicity of tan. 成功解出三角函数方程，特别注意tan的周期性。
Correctly adjust the solution range when the variable inside the trigonometric function is multiplied by a constant (e.g., sin(2x)). 当三角函数内部变量被常数相乘时（如sin(2x)），能正确调整解域范围。
Apply core trigonometric identities (Pythagorean) to simplify and prove identities. 应用核心三角恒等式（毕达哥拉斯恒等式）来简化和证明恒等式。

2. Course Content Overview 2. 课程内容概览

Main Teaching Activities and Time Allocation 主要教学活动和时间分配

Recap: Solving tan equation: Recapping solving 2tan(theta) - 15 degrees = 3.7 for solutions between -180 and 180, focusing on periodicity (adding/subtracting 180).

复习：求解tan方程： 复习求解-180到180之间 2tan(theta) - 15度 = 3.7的解，重点关注周期性（加/减180度）。

New Concept: Range adjustment for sin(2x): Solving sin(2*theta) = 0.3, emphasizing the necessity of finding all possible values for 2*theta within the broadened range (-360 to 360) *before* dividing by two.

新概念：sin(2x)方程的范围调整： 求解sin(2*theta) = 0.3，强调必须在*除以二之前*，在拓宽的范围内（-360到360）找出2*theta的所有可能值。

Revision: Algebraic Trigonometric Equations: Revisiting algebraic manipulation of identities, focusing on substitution to create quadratic equations in terms of one trig function (e.g., Q2b).

复习：代数三角方程： 复习恒等式的代数运算，重点关注通过替换创建关于一个三角函数的二次方程（例如Q2b）。

Application: Proving Identities: Working through identity proofs (Q5a, 5b), emphasizing the use of sin^2x + cos^2x = 1 to convert equations to a single trig function quadratic form.

应用：证明恒等式： 完成恒等式证明（Q5a, 5b），强调使用sin^2x + cos^2x = 1将方程转化为单一三角函数的二次形式。

Advanced Proof Technique & Next Steps: Tackling a complex proof (Q5c) using algebraic manipulation (Difference of Two Squares) and previewing reciprocal functions (cot, sec, cosec).

高级证明技巧与后续步骤： 处理一个复杂的证明（Q5c），使用代数技巧（平方差）并预览倒数函数（cot, sec, cosec）。

Language Knowledge and Skills 语言知识与技能

Vocabulary:
Asymptotes, Periodicity, Inverse Sine (arcsin), Inverse Tangent (arctan), Pythagorean Identity, Quadratic, Reciprocal Functions (cotangent, secant, cosecant).

词汇：
渐近线, 周期性, 反正弦（arcsin）, 反正切（arctan）, 毕达哥拉斯恒等式, 二次方程, 倒数函数（余切、正割、余割）。

Concepts:
Trigonometric Periodicity (tan = 180 degrees), Range Broadening for Composite Functions, Converting equations to quadratic form using sin^2(x) = 1 - cos^2(x), Proof strategies (start one side, use algebraic/identity tools, aim for the other side).

概念：
三角函数周期性（tan为180度）, 复合函数的范围拓宽, 使用sin^2(x) = 1 - cos^2(x)将方程转化为二次形式, 证明策略（从一边开始，使用代数/恒等式工具，目标是另一边）。

Skills Practiced:
Solving trigonometric equations, algebraic manipulation, applying fundamental trigonometric identities, graphical interpretation of solutions, step-by-step proof construction.

练习技能：
解三角函数方程, 代数运算, 应用基本三角恒等式, 解的图形解释, 逐步构建证明。

Teaching Resources and Materials 教学资源与材料

Worksheet with various trigonometric equations (tan, sin 2x forms). 包含各种三角函数方程（tan, sin 2x形式）的练习纸。
Worksheet with trigonometric identity proofs (Q5). 包含三角恒等式证明的练习纸（Q5）。

3. Student Performance Assessment (Henry) 3. 学生表现评估 (Henry)

Participation and Activeness 参与度和积极性

Excellent sustained focus throughout the session, actively engaging with both recap and new material. 在整个课程中保持了优秀的专注度，积极参与了复习和新知识的学习。

Language Comprehension and Mastery 语言理解和掌握

Demonstrated strong understanding of tan periodicity. Initially needed guidance on adjusting ranges for sin(2x) but grasped the concept quickly after explanation. 展示了对tan周期性的深刻理解。最初在调整sin(2x)的范围时需要指导，但在解释后迅速掌握了概念。
Showed good ability to select and apply the correct identity (e.g., converting to a quadratic) during proof work. 在证明练习中，展示了选择和应用正确恒等式（例如转化为二次方程）的良好能力。

Language Output Ability 语言输出能力

Oral: 口语：

Clear articulation when describing steps, although occasional self-correction (e.g., signs or required steps) suggests thinking ahead of writing. 在描述步骤时表达清晰，但偶尔的自我修正（例如符号或所需步骤）表明思考速度快于书写速度。

Written: 书面：

Work shown for most complex steps was accurate after minor correction/prompting. Graph sketching for solution verification was effective.

在经过轻微指导或提示后，大多数复杂步骤的笔录是准确的。用于验证解的图形绘制非常有效。

Student's Strengths 学生的优势

Strong grasp of algebraic substitution to create quadratic equations from trigonometric expressions. 对代数替换以从三角表达式创建二次方程有很强的把握。
Quickly picked up the subtle rule regarding range adjustment for composite functions (sin(2x)). 快速掌握了复合函数（sin(2x)）范围调整的微妙规则。
Good intuition for applying the Pythagorean identity in proof work. 在证明工作中应用毕达哥拉斯恒等式方面有良好的直觉。

Areas for Improvement 需要改进的方面

Needs practice in systematically listing all required steps in complex proofs without rushing or skipping logical connections. 需要在复杂的证明中系统地列出所有必需的步骤，避免仓促或跳过逻辑联系。
Ensure all boundary conditions (like the interval range) are explicitly checked against the transformation of the variable (e.g., 2*theta). 确保所有边界条件（如区间范围）都与变量的变换（例如2*theta）进行明确的对照检查。

4. Teaching Reflection 4. 教学反思

Effectiveness of Teaching Methods 教学方法的有效性

Highly effective, using the recap and comparison method (tan vs sin 2x) to introduce complex range adjustment concepts. 非常有效，通过复习和对比方法（tan vs sin 2x）引入复杂的范围调整概念。
The focus on 'why' certain steps are taken (like substituting identities before dividing) built deeper conceptual understanding. 专注于解释采取某些步骤的“原因”（例如在除法前代入恒等式）培养了更深层次的理解。

Teaching Pace and Time Management 教学节奏和时间管理

The pace was well managed, starting slow for recap and accelerating appropriately during the dense identity proof section. 节奏管理得当，从复习部分开始放缓，在密集的恒等式证明部分适当加速。

Classroom Interaction and Atmosphere 课堂互动和氛围

Engaged, focused, and interactive. The student responded well to challenges and detailed explanations.

专注、投入且互动性强。学生对挑战和详细解释反应良好。

Achievement of Teaching Objectives 教学目标的达成

All objectives were substantially met, with strong progression shown in handling composite trig functions and identity manipulation. 所有目标都基本达成，在处理复合三角函数和恒等式操作方面显示出强劲的进步。

5. Subsequent Teaching Suggestions 5. 后续教学建议

Teaching Strengths 教学优势

Identified Strengths: 识别的优势：

Effective scaffolding: building from known (tan periodicity) to unknown (sin 2x range adjustment). 有效的脚手架：从已知（tan周期性）构建到未知（sin 2x 范围调整）。
Clear explanation of the rationale behind placing the range adjustment step at the inverse function application point. 清晰解释了将范围调整步骤放在反函数应用点的背后的原理。

Effective Methods: 有效方法：

Encouraging sketching graphs (for sin) to visualize required solutions and symmetry. 鼓励绘制图形（用于sin函数）以可视化所需的解和对称性。
Guiding student through algebraic proof strategies by focusing on the target form and appropriate identity choice. 通过关注目标形式和适当的恒等式选择，指导学生完成代数证明策略。

Positive Feedback: 正面反馈：

Excellent spotting of the 'Difference of Two Squares' factorization during the complex proof. 在复杂的证明中，出色地发现了“平方差”的因式分解。

Next Teaching Focus 下一步教学重点

Deepening understanding of reciprocal trigonometric functions (cotangent, secant, cosecant) and their related identities. 深化对倒数三角函数（余切、正割、余割）及其相关恒等式的理解。

Specific Suggestions for Student's Needs 针对学生需求的具体建议

Equation Solving: 方程求解：

Always explicitly state the adjusted range for the internal angle before solving for the variable when dealing with equations like sin(2x) or cos(3x). 在处理sin(2x)或cos(3x)这类方程时，在求解内部角度前，务必明确写出调整后的范围。

Proof Writing: 证明书写：

In proofs, state the identity you are using explicitly before substitution to ensure logical clarity, especially when combining multiple identities. 在证明中，代入前明确写出所使用的恒等式，以确保逻辑清晰，尤其是在组合多个恒等式时。

Recommended Supplementary Learning Resources or Homework 推荐的补充学习资源或家庭作业

Complete the remaining algebraic identity proofs (Q5c, d, etc.) on the provided sheet. 完成所提供练习纸上剩余的代数恒等式证明题（Q5c, d等）。
Review resources on Physics & Maths Tutor for practice on reciprocal functions. 查阅Physics & Maths Tutor上的资源，练习倒数函数的解题。