Trigonometric Identities and Equations | 三角恒等式与方程

📚 Trigonometric Identities and Equations | 三角恒等式与方程

In Edexcel A Level Mathematics, trigonometric identities and equations are central to Pure Mathematics. You are expected to move beyond right-angled triangles, work in both degrees and radians, and solve equations over given intervals. This guide covers the key identities, exact values, solution strategies, and common pitfalls.

在 Edexcel A Level 数学中，三角恒等式与方程是纯数学的核心内容。你需要超越直角三角形，同时使用角度制和弧度制，并在指定区间内解方程。本指南涵盖关键恒等式、特殊角精确值、解题策略和常见陷阱。

1. The Core Identities: sin² θ, cos² θ, and tan θ | 核心恒等式：sin² θ、cos² θ 与 tan θ

In Edexcel Pure Year 1, the two essential identities are sin² θ + cos² θ ≡ 1 and tan θ ≡ sin θ / cos θ.

在 Edexcel 纯数学第一年，两个基本恒等式是 sin² θ + cos² θ ≡ 1 和 tan θ ≡ sin θ / cos θ。

The first identity is called the Pythagorean identity because it follows from the unit circle or Pythagoras’ theorem.

第一个恒等式称为毕达哥拉斯恒等式，因为它源于单位圆或毕达哥拉斯定理。

The quotient identity tan θ ≡ sin θ / cos θ is valid whenever cos θ ≠ 0.

商恒等式 tan θ ≡ sin θ / cos θ 在 cos θ ≠ 0 时成立。

sin² θ + cos² θ ≡ 1

tan θ ≡ sin θ / cos θ

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2. Extending to Reciprocal Identities | 扩展到倒数恒等式

In Year 2, you also need secant, cosecant, and cotangent: sec θ ≡ 1 / cos θ, cosec θ ≡ 1 / sin θ, and cot θ ≡ cos θ / sin θ.

第二年还需掌握正割、余割和余切：sec θ ≡ 1 / cos θ，cosec θ ≡ 1 / sin θ，cot θ ≡ cos θ / sin θ。

These lead to two further Pythagorean forms: 1 + tan² θ ≡ sec² θ and 1 + cot² θ ≡ cosec² θ.

它们导出另外两个毕达哥拉斯形式：1 + tan² θ ≡ sec² θ 和 1 + cot² θ ≡ cosec² θ。

They are useful when simplifying expressions or proving more advanced identities.

它们在化简表达式或证明更复杂的恒等式时非常有用。

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

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3. Exact Values You Must Know | 必须熟记的精确值

Exact values for 0°, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2 radians) are frequently required. You should not need a calculator for these.

0°、30°、45°、60°、90°（或 0、π/6、π/4、π/3、π/2 弧度）的精确值经常被考查，你不应该使用计算器。

θ (deg)	θ (rad)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3 = √3/3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

You can extend these values to other quadrants using symmetry and the CAST diagram.

你可以利用对称性和 CAST 图将这些值推广到其他象限。

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4. CAST Diagram and Signs in Quadrants | CAST 图与象限符号

The CAST diagram tells you which trigonometric ratios are positive in each quadrant: All in QI, Sin in QII, Tan in QIII, Cos in QIV.

CAST 图告诉你各象限中哪些三角比为正：第一象限全部为正，第二象限 sin 为正，第三象限 tan 为正，第四象限 cos 为正。

This is essential for finding all solutions between 0° and 360° or 0 and 2π.

这对于找出 0° 到 360° 或 0 到 2π 之间的所有解至关重要。

Always sketch the relevant quadrant before listing solutions.

在列出解之前，始终画出相关象限。

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5. Solving Basic Trigonometric Equations | 解基本三角方程

To solve sin θ = 0.5 for 0° ≤ θ ≤ 360°, first find the principal value θ = sin⁻¹(0.5) = 30°.

解 sin θ = 0.5 在 0° ≤ θ ≤ 360° 时，先找主值 θ = sin⁻¹(0.5) = 30°。

Since sin is positive in QI and QII, the second solution is 180° – 30° = 150°.

由于 sin 在第一、二象限为正，第二个解是 180° – 30° = 150°。

The solutions are θ = 30°, 150°.

因此解为 θ = 30°、150°。

For cos θ = -0.5, the principal value is 120°, and the other solution is 240° because cos is negative in QII and QIII.

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