Differentiating sin x and cos x | 正弦与余弦函数的微分

📚 Differentiating sin x and cos x | 正弦与余弦函数的微分

In A-Level Edexcel Mathematics, differentiating trigonometric functions is a core skill that appears across pure mathematics, mechanics and applied contexts. This article focuses on the two most fundamental results: the derivatives of sin x and cos x. You need to know the standard results, understand where they come from, and be able to apply the chain rule, product rule and quotient rule confidently.

在 A-Level Edexcel 数学中，对三角函数求导是一项核心技能，贯穿纯数学、力学和应用情境。本文重点讨论两个最基本的结果：sin x 和 cos x 的导数。你需要掌握标准结论，理解它们的来源，并能够熟练应用链式法则、乘积法则和商法则。

1. The Key Results | 核心结论

The two standard derivatives you must memorise are given below. They are the foundation for nearly all differentiation involving trigonometric functions at A-Level.

以下是两个必须牢记的标准导数结果。它们是 A-Level 中几乎所有涉及三角函数的微分问题的基础。

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

These results only apply when x is measured in radians, not degrees. For a constant multiple, the derivative scales in the usual way: if y = 5 sin x, then dy/dx = 5 cos x, and if y = -2 cos x, then dy/dx = 2 sin x.

这些结果仅在 x 使用弧度制时成立，而不是度数制。对于常数倍，导数按通常规则缩放：若 y = 5 sin x，则 dy/dx = 5 cos x；若 y = -2 cos x，则 dy/dx = 2 sin x。

2. Why Radian Measure Matters | 为什么必须使用弧度制

At A-Level, trigonometric differentiation is always assumed to be in radians unless a question explicitly states otherwise. The reason comes from the small-angle limit that underpins the first-principles proof.

在 A-Level 考试中，除非题目明确说明，否则三角函数的微分均默认使用弧度制。原因在于支撑第一原理证明的小角度极限。

lim (h → 0) sin h / h = 1

This limit only equals 1 when h is measured in radians. If h were in degrees, the limit would involve an extra factor of π/180, and the derivative of sin x would become (π/180) cos x. Exam questions assume radians, so do not add this factor unless told otherwise.

这个极限只有在 h 以弧度表示时才等于 1。如果 h 以度数表示，极限就会含有一个额外的因子 π/180，sin x 的导数也会变成 (π/180) cos x。考试题目默认使用弧度制，因此除非题目另作说明，否则不要加上这个因子。

3. Derivative of sin x from First Principles | 从第一原理推导 sin x 的导数

By definition,

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