📚 Exercise 21A: Trigonometric Identities | 练习21A：三角恒等式
Trigonometric identities are equations involving trigonometric functions that hold true for all permissible values of the variable. In IB Mathematics, mastering these identities is essential for simplifying complex expressions, solving equations, and proving mathematical statements. Exercise 21A typically focuses on building a strong foundation with fundamental identities, including Pythagorean, reciprocal, and quotient relationships, while also extending to compound-angle and double-angle formulas. This article provides a step-by-step guide to the core concepts, problem-solving strategies, and common pitfalls, ensuring you gain confidence for your exams.
三角恒等式是涉及三角函数的等式，它们在变量允许取值的范围内恒成立。在 IB 数学中，掌握这些恒等式对于化简复杂表达式、求解方程和证明数学命题至关重要。练习 21A 通常注重通过毕达哥拉斯恒等式、倒数恒等式和商数恒等式打下坚实基础，同时延伸至和差角公式与倍角公式。本文提供核心概念、解题策略和常见误区的逐步指导，帮助你在考试中建立信心。
1. Understanding Trigonometric Identities | 理解三角恒等式
relationships. For instance, the identity sin²θ + cos²θ = 1 can be used to replace sin²θ with 1 − cos²θ, unlocking simpler algebraic manipulation. In IB Maths, recognising when and how to apply an identity is a key assessment objective, and Exercise 21A systematically builds this skill.
恒等式是一个对所有有效输入都成立的等式，而普通方程仅在特定值上满足。在三角学中，恒等式允许我们将表达式改写成等价形式，这对于化简复杂函数或证明关系至关重要。例如，恒等式 sin²θ + cos²θ = 1 可用于将 sin²θ 替换为 1 − cos²θ，从而开启更简单的代数运算。在 IB 数学中，识别何时以及如何应用恒等式是一项关键的评估目标，练习 21A 系统地培养这一技能。
2. The Pythagorean Identity | 毕达哥拉斯恒等式
sc²θ. These three versions are indispensable when you need to convert between sine, cosine, tangent, and their reciprocals, especially in proving more advanced identities.
从单位圆推导而来，单位圆上的点 (x, y) 满足 x² + y² = 1，且 x = cos θ, y = sin θ，由此得到基本的毕达哥拉斯恒等式 sin²θ + cos²θ = 1。将此恒等式除以 cos²θ 或 sin²θ，可以得到另外两种常用形式：1 + tan²θ = sec²θ 和 1 + cot²θ = csc²θ。当需要在正弦、余弦、正切及其倒数之间转换时，这三个版本是不可或缺的，尤其是在证明更复杂的恒等式时。
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
3. Reciprocal Identities | 倒数恒等式
hat greatly simplifies simplification and proof tasks. Always be mindful of domain restrictions: for example, csc θ is undefined when sin θ = 0.
倒数恒等式用正弦、余弦和正切定义了余割、正割和余切函数：csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。在练习 21A 早期，它们常被用于将一个表达式中的所有三角函数重写为正弦和余弦的形式，这一策略能极大地简化化简和证明任务。务必留意定义域的限制：例如，当 sin θ = 0 时，csc θ 无定义。
csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
4. Quotient Identities | 商数恒等式
bly the most frequently used identities after the Pythagorean ones. Whenever you encounter tan θ or cot θ in an expression that also involves sin θ or cos θ, converting to sines and cosines will often reveal cancellations and hidden Pythagorean structures.
商数恒等式将正切和余切直接表示为正弦与余弦的比值：tan θ = sin θ / cos θ，cot θ = cos θ / sin θ。这可以说是继毕达哥拉斯恒等式之后最常用的恒等式。每当在同时包含 sin θ 或 cos θ 的表达式中遇到 tan θ 或 cot θ 时，将它们转换为正弦和余弦往往会揭示出可约分项和隐藏的毕达哥拉斯结构。
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
5. Simplifying Trigonometric Expressions | 化简三角表达式
ncel common factors. For example, simplify (csc θ − sin θ) / cot θ. Rewriting gives (1/sin θ − sin θ) ÷ (cos θ/sin θ) = ((1 − sin²θ)/sin θ) × (sin θ/cos θ) = cos²θ / (sin θ) × (sin θ/cos θ) = cos θ.
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