📚 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)。
Radian measure is defined as the ratio of the arc length to the radius: θ = s / 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。
2. Graphs of Trigonometric Functions | 三角函数的图像
The graph of y = sin x is a wave with period 2π, amplitude 1, and range [-1, 1]. The 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。
The graph of y = tan x has asymptotes 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。
Transformations 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。
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。
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。
These formulae are essential for deriving double-angle formulae, 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。
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
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply