Using Trigonometric Identities | 运用三角恒等式

📚 Using Trigonometric Identities | 运用三角恒等式

Trigonometric identities are fundamental tools for simplifying expressions, solving equations, and proving relationships in A-Level Mathematics. Mastering these identities not only boosts your algebraic manipulation skills but also lays a solid groundwork for calculus and further study. This article covers essential identities such as Pythagorean, double-angle, compound-angle, and the R-formula, along with practical techniques for using them in exam-style problems.

三角恒等式是简化表达式、解方程和证明关系的核心工具，在 A-Level 数学中至关重要。熟练掌握这些恒等式不仅能提高代数运算能力，还为微积分和进阶学习打下坚实基础。本文将涵盖基本恒等式，如勾股恒等式、倍角公式、和差角公式以及 R 公式，同时介绍如何在考试题型中灵活运用它们。

1. Fundamental Pythagorean Identities | 基本勾股恒等式

The three Pythagorean identities are derived from the unit circle and are essential for simplifying trigonometric expressions. The first identity is sin²θ + cos²θ ≡ 1. Dividing through by cos²θ gives 1 + tan²θ ≡ sec²θ, and dividing by sin²θ yields 1 + cot²θ ≡ cosec²θ.

三个基本勾股恒等式源自单位圆，是化简三角表达式的关键。第一个恒等式为 sin²θ + cos²θ ≡ 1。将其两边除以 cos²θ 可得 1 + tan²θ ≡ sec²θ，除以 sin²θ 可得 1 + cot²θ ≡ cosec²θ。

sin²θ + cos²θ ≡ 1

1 + tan²θ ≡ sec²θ

1 + cot²θ ≡ cosec²θ

These identities hold for all values of θ where the functions are defined. They are particularly useful when you need to replace one trigonometric function with another, for instance converting between sin²θ and cos²θ.

这些恒等式对所有使函数有定义的 θ 都成立。当你需要用一个三角函数替换另一个时，它们特别有用，例如在 sin²θ 和 cos²θ 之间进行转换。

Example: Simplify 2 sin²θ + 2 cos²θ. Using the identity, sin²θ + cos²θ = 1, the expression simplifies to 2(1) = 2.

示例：化简 2 sin²θ + 2 cos²θ。利用恒等式 sin²θ + cos²θ = 1，表达式简化为 2 × 1 = 2。

2. Reciprocal and Ratio Identities | 倒数与比值恒等式

The reciprocal identities define sec, cosec, and cot in terms of cos, sin, and tan: secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = 1/tanθ = cosθ/sinθ.

倒数恒等式用 cos、sin 和 tan 定义了 sec、cosec 和 cot：secθ = 1

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