A-Level Mathematics: Trigonometric Functions and Identities | A-Level 数学：三角函数与恒等式

📚 A-Level Mathematics: Trigonometric Functions and Identities | A-Level 数学：三角函数与恒等式

Trigonometry is one of the most heavily examined topics in CIE A-Level Mathematics, appearing in both Pure Mathematics 1 (P1) and Pure Mathematics 3 (P3). A solid grasp of trigonometric functions and identities is essential for solving equations, proving statements, and modelling periodic phenomena.

三角函数与恒等式是 CIE A-Level 数学中考查频率极高的内容，在纯数学 1（P1）与纯数学 3（P3）中均占据重要地位。扎实掌握三角函数与恒等式，对于解方程、证明恒等式以及模拟周期现象都至关重要。

1. The Unit Circle and Radian Measure | 单位圆与弧度制

The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured anticlockwise from the positive x-axis, the coordinates of the point where the terminal side meets the circle are (cos θ, sin θ).

单位圆是圆心在原点、半径为 1 的圆。对于从 x 轴正方向逆时针量取的任意角 θ，终边与单位圆交点的坐标为 (cos θ, sin θ)。

r. A full revolution equals 2π radians, which corresponds to 360°. Key conversions to memorise:

弧度制定义为弧长与半径之比：θ = s / r。一整圈等于 2π 弧度，对应 360°。需要牢记的关键换算：

• 180° = π radians | 180° = π 弧度
• 90° = π/2, 60° = π/3, 45° = π/4, 30° = π/6

On the unit circle, sin θ is the y-coordinate and cos θ is the x-coordinate. This geometric view explains why sin θ is positive in quadrants I and II, while cos θ is positive in quadrants I and IV.

在单位圆上，sin θ 是纵坐标，cos θ 是横坐标。这一几何视角解释了为什么 sin θ 在第一、二象限为正，而 cos θ 在第一、四象限为正。

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2. Graphs of Trigonometric Functions | 三角函数的图像

graph of y = cos x is a horizontal translation of the sine graph by π/2 units to the left.

y = sin x 的图像是周期为 2π、振幅为 1、值域为 [-1, 1] 的波形。y = cos x 的图像是正弦图像向左平移 π/2 个单位的结果。

s at x = π/2 + nπ (where n is an integer), a period of π, and a range of all real numbers.

y = tan x 的图像在 x = π/2 + nπ（n 为整数）处有渐近线，周期为 π，值域为全体实数。

ons of trigonometric graphs follow the standard rules: y = a sin(bx + c) + d has amplitude |a|, period 2π/|b|, phase shift -c/b, and vertical shift d.

三角函数图像的变换遵循标准规则：y = a sin(bx + c) + d 的振幅为 |a|，周期为 2π/|b|，相位偏移为 -c/b，垂直平移为 d。

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3. Basic Trigonometric Identities | 基本三角恒等式

For CIE A-Level, the following basic identities must be memorised and applied fluently:

对于 CIE A-Level，以下基本恒等式必须熟练记忆并灵活运用：

• tan θ ≡ sin θ / cos θ
  tan θ ≡ sin θ / cos θ
• sin²θ + cos²θ ≡ 1
  sin²θ + cos²θ ≡ 1
• 1 + tan²θ ≡ sec²θ
  1 + tan²θ ≡ sec²θ
• 1 + cot²θ ≡ csc²θ
  1 + cot²θ ≡ csc²θ

The identity sin²θ + cos²θ ≡ 1 follows directly from the Pythagorean theorem applied to the unit circle. Dividing it by cos²θ yields 1 + tan²θ ≡ sec²θ, and dividing by sin²θ yields 1 + cot²θ ≡ csc²θ.

恒等式 sin²θ + cos²θ ≡ 1 直接来自勾股定理在单位圆上的应用。将其除以 cos²θ 可得 1 + tan²θ ≡ sec²θ，除以 sin²θ 可得 1 + cot²θ ≡ csc²θ。

These identities are used to simplify expressions, prove other identities, and convert trigonometric equations into forms that are easier to solve.

这些恒等式用于化简表达式、证明其他恒等式，以及将三角方程转化为更易于求解的形式。

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4. Compound Angle Formulae | 复合角公式

The compound angle formulae expand sine, cosine, and tangent of sums and differences:

复合角公式将和差角的正弦、余弦和正切展开：

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

A common mnemonic: "Sine keeps the same sign, cosine changes the sign." In the sine formula the sign is preserved; in the cosine formula the sign is reversed.

一个常用口诀："正弦同号，余弦变号。"在正弦公式中符号保持一致；在余弦公式中符号取反。

mulae, solving equations such as sin 2θ = cos θ, and evaluating exact values like sin 75°.

这些公式是推导二倍角公式、求解如 sin 2θ = cos θ 的方程以及计算 sin 75° 等精确值的基础。

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5. Double Angle and Half Angle Formulae | 二倍角与半角公式

Setting B = A in the compound angle formulae yields the double

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