📚 Small Angle Approximations | 小角度近似
In A-Level Mathematics, small angle approximations allow you to replace sin θ, cos θ and tan θ with simple polynomial expressions when θ is measured in radians and is close to zero. These formulas are extremely useful in mechanics, wave theory and numerical estimation, and they follow directly from Maclaurin series.
在 A-Level 数学中，小角度近似允许你在 θ 以弧度为单位且接近零时，用简单的多项式表达式代替 sin θ、cos θ 和 tan θ。这些公式在力学、波动理论和数值估算中非常有用，它们直接来自麦克劳林级数。
1. Radian Measure and the Meaning of “Small” | 弧度制与”小”的含义
Small angle approximations are only valid when θ is measured in radians, not degrees. This is because the derivative limits such as sin θ / θ → 1 as θ → 0 only hold in radian measure. If θ is given in degrees, you must first convert to radians using θ_rad = θ_deg × π / 180.
小角度近似仅在 θ 以弧度为单位时成立，而不是度数。这是因为诸如当 θ → 0 时 sin θ / θ → 1 的导数极限只在弧度制下成立。如果 θ 给的是度数，你必须先用 θ_rad = θ_deg × π / 180 换算成弧度。
There is no universal cut-off for “small”, but Edexcel exam questions usually use angles below about 0.1 radians, roughly 5.7 degrees, unless a stated error tolerance says otherwise. At 0.1 radians, sin θ ≈ θ is already accurate to about 0.17%.
“小”没有绝对的截止点，但 Edexcel 试题通常使用小于约 0.1 弧度的角度，大约是 5.7 度，除非题目给出了特定的误差容限。在 0.1 弧度时，sin θ ≈ θ 已经精确到约 0.17%。
2. The Standard Approximations | 标准近似公式
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