Double Angle Formulae for Edexcel A Level | Edexcel A Level 二倍角公式精讲

📚 Double Angle Formulae for Edexcel A Level | Edexcel A Level 二倍角公式精讲

The double angle formulas are a cornerstone of A Level trigonometry, enabling the simplification of expressions, solving of complex equations, and providing a bridge to calculus techniques. In this revision guide tailored for Edexcel Mathematics, we derive each formula, illustrate their applications, and highlight typical pitfalls to ensure mastery.

二倍角公式是 A Level 三角学的基石，能够化简表达式、求解复杂方程，并为微积分技巧搭建桥梁。在这份为 Edexcel 数学量身定制的复习指南中，我们推导每一个公式，展示其应用，并指出常见误区，确保牢固掌握。

1. Recap of Basic Trigonometric Identities | 基本三角恒等式回顾

addition formulas sin(A + B) = sinA cosB + cosA sinB, cos(A + B) = cosA cosB − sinA sinB.

在探索二倍角之前，必须回顾基本恒等式：sin²θ + cos²θ = 1，tanθ = sinθ / cosθ，以及加法公式 sin(A + B) = sinA cosB + cosA sinB，cos(A + B) = cosA cosB − sinA sinB。

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2. Deriving the Double Angle Formula for Sine | 正弦二倍角公式推导

y is the most straightforward to remember and is frequently used to rewrite products as sums.

在正弦加法公式中令 A = B = θ，得到 sin(θ + θ) = sinθ cosθ + cosθ sinθ，直接简化为 sin 2θ = 2 sinθ cosθ。这个恒等式最易记忆，常用于将乘积改写为和的形式。

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3. Deriving the Double Angle Formula for Cosine | 余弦二倍角公式推导

s²θ − 1 and cos 2θ = 1 − 2 sin²θ. Each variant has its own advantage depending on the context.

将 A = B = θ 代入 cos(A + B) 得到 cos 2θ = cosθ cosθ − sinθ sinθ = cos²θ − sin²θ。运用 sin²θ + cos²θ = 1，可得到两种等价形式：cos 2θ = 2 cos²θ − 1 和 cos 2θ = 1 − 2 sin²θ。不同形式在不同情境下各有优势。

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4. Deriving the Double Angle Formula for Tangent | 正切二倍角公式推导

y cos²θ, provided cosθ ≠ 0, leads to tan 2θ = (2 tanθ) / (1 − tan²θ). This formula is particularly useful when working with tangent-only equations.

由 tan 2θ = sin 2θ / cos 2θ 及二倍角表达式，可写为 tan 2θ = (2 sinθ cosθ) / (cos²θ − sin²θ)。分子分母同除以 cos²θ（cosθ ≠ 0），即得 tan 2θ = (2 tanθ) / (1 − tan²θ)。该公式在仅含正切的方程中尤其有用。

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5. Simplifying Expressions Using Double Angle Identities | 运用二倍角恒等式化简表达式

ilarly, 6 cos²x − 3 becomes 3 cos 2x after applying cos 2x = 2 cos²x − 1 and a little algebraic manipulation.

二倍角公式能将复杂的三角表达式化简。比如，8 sinx cosx 可表示为 4 sin 2x。类似地，6 cos²x − 3 在应用 cos 2x = 2 cos²x − 1 并稍作代数处理后可化为 3 cos 2x。

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6. Solving Trigonometric Equations with Double Angles | 用二倍角求解三角方程

a single angle. For instance, sin 2x = cos x can be rewritten as 2 sinx cosx = cos x. Factorising yields cos x (2 sin x − 1) = 0, leading to cos x = 0 or sin x = ½.

当方程中同时含有 sin 2θ、cos 2θ 或 tan 2θ 与单角项时，代入二倍角恒等式可将方程化为单一角度。例如 sin 2x = cos x 可改写为 2 sinx cosx = cos x。因式分解得 cos x (2 sin x − 1) = 0，从而 cos x = 0 或 sin x = ½。

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7. Solving Equations with cos 2θ Variants | 运用余弦二倍角变形求解

ing cos 2θ = 1 − 2 sin²θ transforms the equation into a quadratic in sinθ: 1 − 2 sin²θ + sinθ = 0, or 2 sin²θ − sinθ − 1 = 0.

许多 Edexcel 考题会出现 cos 2θ 与平方项同时存在的情况。选择合适的 cos 2θ 形式至关重要。对于 cos 2θ + sinθ = 0，采用 cos 2θ = 1 − 2 sin²θ 可将方程转化为关于 sinθ 的二次方程：1 − 2 sin²θ + sinθ = 0，即 2 sin²θ − sinθ − 1 = 0。

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8. Proving Identities Involving Double Angles | 证明含二倍角的恒等式

t-hand side, substitute cos 2θ = 1 − 2 sin²θ in the numerator and cos 2θ = 2 cos²θ − 1 in the denominator to obtain (2 sin²θ) / (2 cos²θ) = tan²θ.

证明题经常出现在 Edexcel 规格中。一个典型例子是证明 (1 − cos 2θ) / (1 + cos 2θ) = tan²θ。从左边开始，分子代入 cos 2θ = 1 − 2 sin²θ，分母代入 cos 2θ = 2 cos²θ − 1，得到 (2 sin²θ) / (2 cos²θ) = tan²θ。

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9. Applications to Integration | 在积分中的应用

= (1 − cos 2θ)/2 enable the integration of sin²x and cos²x. For example, ∫ sin²x dx = ∫ ½(1 − cos 2x) dx = ½x − ¼ sin 2x + C.

二倍角公式在积分中不可或缺，常常反向使用以降低幂次。恒等式 cos²θ = (1 + cos 2θ)/2 和 sin²θ = (1 − cos 2θ)/2 使得 sin²x 和 cos²x 的积分成为可能。例如，∫ sin²x dx = ∫ ½(1 − cos 2x) dx = ½x − ¼ sin 2x + C。

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10. Worked Examples and Examination Practice | 例题与考试演练

Example: Solve cos 2x = sin x for 0 ≤ x ≤ 2π.
sin x − 1)(sin x + 1) = 0. Hence sin x = ½ or sin x = −1. Solutions: x = π/6, 5π/6, 3π/2.

例题：解方程 cos 2x = sin x，0 ≤ x ≤ 2π。
代入 cos 2x = 1 − 2 sin²x 得 1 − 2 sin²x = sin x → 2 sin²x + sin x − 1 = 0 → (2 sin x − 1)(sin x + 1) = 0。故 sin x = ½ 或 sin x = −1。解为 x = π/6、5π/6、3π/2。

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11. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Students often misremember cos 2θ as cos²θ + sin²θ or forget the factor of 2 in sin 2θ. Another frequent error is failing to consider the quadrant when finding angles from trigonometric values. Always double-check the range and include all possible solutions within the given interval.

学生常将 cos 2θ 误记为 cos²θ + sin²θ，或漏掉 sin 2θ 中的因子 2。另一个常见错误是由三角值求角度时忽略象限。务必检查范围，并包含给定区间内的所有可能解。

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12. Summary and Key Takeaways | 总结与关键要点

Master the three double angle forms for cosine, ensure you can rapidly switch between them, and practice applying identities to both equation solving and integration. Memorise the derivative forms for integration, and always express final solutions in the simplest exact form required by Edexcel mark schemes.

熟练掌握余弦的三种二倍角形式，确保能在它们之间快速转换，并多加练习将恒等式应用于解方程和积分。牢记用于积分的变形公式，并始终按 Edexcel 评分方案的要求，将最终答案表示为最简精确形式。

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