📚 Small Angle Approximations | 小角度近似
In A-level Mathematics, especially within the Edexcel specification, small angle approximations provide a powerful technique for simplifying trigonometric expressions when the angle is measured in radians and is close to zero. By replacing sin θ, cos θ and tan θ with polynomial approximations, we can evaluate limits, solve equations and analyse physical situations more easily. These approximations stem from the Maclaurin series expansions of the trigonometric functions and are remarkably accurate for angles with small magnitude.
在A-level数学中，特别是Edexcel考试大纲中，小角度近似为求解三角函数问题提供了一种强有力的简化方法。当角度以弧度为单位且趋近于零时，我们用多项式近似代替 sin θ、cos θ 和 tan θ，从而更便捷地计算极限、解方程以及分析物理情景。这些近似源自三角函数的麦克劳林级数展开，对于幅度很小的角度具有极高的精确度。
1. The Need for Radian Measure | 使用弧度制的必要性
The small angle approximations are only valid when the angle is expressed in radians. This is because the Maclaurin series expansion for sin θ, cos θ and tan θ rely on the radian definition of trigonometric functions. If an angle is given in degrees, you must first convert it to radians by multiplying by π/180 before applying any approximation.
小角度近似仅在角度以弧度表示时才成立。这是因为 sin θ、cos θ 和 tan θ 的麦克劳林级数展开依赖于三角函数的弧度定义。如果题目给出的角度是度数，必须先用 π/180 相乘转换为弧度，再应用近似公式。
because the degree measure is not directly proportional to the radian measure in the series derivation.
例如，若要近似 sin 3°，应使用 θ = 3 × π/180 = π/60 弧度，然后 sin θ ≈ θ = π/60。如果尝试用 sin 3° ≈ 3，那将是完全错误的，因为在级数推导中角度数值必须基于弧度，而非度数。
2. Derivation from Maclaurin Series | 由麦克劳林级数推导
The Maclaurin series for sin θ is sin θ = θ − θ³/3! + θ⁵/5! − θ⁷/7! + ... . When |θ| the simplest approximation sin θ ≈ θ. Similarly, the series for cos θ is cos θ = 1 − θ²/2! + θ⁴/4! − θ⁶/6! cos θ ≈ 1, but usually we retain the θ² term to achieve better accuracy: cos θ ≈ 1 − θ²/2.
sin θ 的麦克劳林级数为 sin θ = θ − θ³/3! + θ⁵/5! − θ⁷/7! + ... 。当 |θ| 很小时，三次及更高次项可以忽略不计，于是得到最简单的近似 sin θ ≈ θ。cos θ 的级数为 cos θ = 1 − θ²/2! + θ⁴/4! − θ⁶/6! + ... 。对于小角 θ，若忽略 θ² 及以上项可得 cos θ ≈ 1，但通常保留 θ² 项以获得更好的精度，即 cos θ ≈ 1 − θ²/2。
s approximation matches the linear term of sine and tangent, which is why both sin θ and tan θ share the same first-order small angle approximation.
tan θ 的麦克劳林级数为 tan θ = θ + θ³/3 + 2θ⁵/15 + ... 。对于非常小的 θ，首项给出 tan θ ≈ θ。这一近似与正弦的一次项相同，因此在小角度下 sin θ 和 tan θ 具有相同的一阶近似。
3. The Three Standard Approximations | 三个标准近似公式
re usually the first-order (or first non-zero term) expansions. Some problems may also require the approximation cos θ ≈ 1, but this is less accurate and rarely used alone; the inclusion of the quadratic term is standard.
总结起来，Edexcel A-level 数学要求掌握的三个小角度近似为：sin θ ≈ θ、cos θ ≈ 1 − θ²/2、tan θ ≈ θ（均为弧度制）。这些通常是展开式的一阶（或首个非零项）近似。有些题目可能也会用到 cos θ ≈ 1，但精度较低，很少单独使用；标准做法是保留二次项。
The following table compares true values with approximations for a few small angles, demonstrating the impressive precision when θ is very small.
下表比较了若干小角度的真实值与近似值，展示了当 θ 极小时近似的惊人精度。
| θ (rad) | sin θ (true) | θ approx | % error sin | cos θ (true) | 1−θ²/2 approx | % error cos |
|---|---|---|---|---|---|---|
| 0.05 | 0.04998 | 0.05 | 0.04% | 0.99875 | 0.99875 | ~0% |
| 0.10 | 0.09983 | 0.1 | 0.17% | 0.99500 | 0.995 | ~0% |
| 0.20 | 0.19867 | 0.2 | 0.67% | 广告 / Sponsored Pearson Edexcel International A Level 培生爱德思 Alevel 考试 Physi… 物理 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥281.0
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