Trigonometric Identities | 三角函数恒等式

📚 Trigonometric Identities | 三角函数恒等式

Trigonometric identities are equations involving sine, cosine, tangent and their reciprocal functions that hold for all permitted values of the variable. In the Edexcel A-Level specification, these identities are used to simplify expressions, prove new results, solve equations and model periodic behaviour.

三角恒等式是包含正弦、余弦、正切及其倒数函数且在变量允许范围内恒成立的等式。在 Edexcel A-Level 考纲中，这些恒等式用于化简表达式、证明新结论、求解方程以及对周期性现象建模。

1. The Pythagorean Identity | 毕达哥拉斯恒等式

For any angle θ, the point (cos θ, sin θ) lies on the unit circle x² + y² = 1. Substituting x = cos θ and y = sin θ gives the most important identity.

对任意角 θ，点 (cos θ, sin θ) 位于单位圆 x² + y² = 1 上。代入 x = cos θ 和 y = sin θ，即可得到最重要的恒等式。

sin² θ + cos² θ ≡ 1

The triple bar ≡ is used because the statement is true for every value of θ, not just for selected solutions. The identity can be rearranged as sin² θ ≡ 1 − cos² θ and cos² θ ≡ 1 − sin² θ, which is useful when converting between sine and cosine.

使用三横线 ≡ 是因为该等式对每个 θ 值都成立，而不仅仅是对某些特定解成立。该恒等式可改写为 sin² θ ≡ 1 − cos² θ 和 cos² θ ≡ 1 − sin² θ，在正弦与余弦之间相互转换时非常有用。

2. Reciprocal and Quotient Identities | 倒数与商数恒等式

Edexcel A-Level also requires confident use of the reciprocal trigonometric functions. They are defined from sin θ, cos θ and tan θ as follows.

Edexcel A-Level 还要求熟练使用倒数三角函数。它们由 sin θ、cos θ 和 tan θ 定义如下。

tan θ ≡ sin θ / cos θ, cot θ ≡ cos θ / sin θ = 1 / tan θ

sec θ ≡ 1 / cos θ, cosec θ ≡ 1 / sin θ

These definitions are not optional extras: they appear in differentiation, integration, trigonometric equations and modelling. Remember that each reciprocal is undefined when the denominator is zero.

这些定义不是可有可无的补充：它们会出现在微分、

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