AS Mathematics: Trigonometric Functions – Key Concepts | AS 数学：三角函数考点精讲

📚 AS Mathematics: Trigonometric Functions – Key Concepts | AS 数学：三角函数考点精讲

Trigonometry is a fundamental branch of mathematics that studies the relationships between angles and side lengths in triangles. In AS Mathematics, trigonometric functions, their graphs, identities, and applications in solving triangles form a significant part of the syllabus. Mastering these concepts is essential for solving equations, modelling periodic phenomena, and progressing to more advanced topics. This revision guide covers key topics including radian measure, the unit circle, special angles, graphs, identities, solving trigonometric equations, the sine and cosine rules, graph transformations, and inverse trigonometric functions.

三角函数是数学中研究三角形中角度与边长关系的重要分支。在 AS 数学课程中，三角函数、其图像、恒等式以及在解三角形中的应用是考纲的重要组成部分。掌握这些概念对于解方程、建模周期现象以及顺利进阶至关重要。本文涵盖弧度制、单位圆、特殊角、图像、恒等式、解三角方程、正弦定理、余弦定理、图像变换以及反三角函数等关键考点。

1. Radians and Angle Conversion | 弧度制与角度转换

A radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius. Since the circumference of a circle is 2πr, a full revolution of 360° corresponds to 2π radians.

弧度定义为圆弧长等于半径时所对的圆心角。因为圆周长为 2πr，所以 360° 完整一周对应 2π 弧度。

To convert from radians to degrees, multiply by 180/π. To convert from degrees to radians, multiply by π/180. These conversion factors are essential when solving equations or interpreting graphs in radian mode.

将弧度转换为度数，乘以 180/π。将度数转换为弧度，则乘以 π/180。这些转换因子在弧度制下解方程或分析图像时不可或缺。

Common conversions you must memorise include: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, and 2π = 360°. Many AS exam questions expect answers in exact radian form unless degrees are specified.

必须牢记的常用转换有：π/6 = 30°，π/4 = 45°，π/3 = 60°，π/2 = 90°，π = 180°，以及 2π = 360°。许多 AS 试题要求以精确弧度值作答，除非题目明确要求用度数。

2. The Unit Circle and Trigonometric Definitions | 单位圆与三角函数定义

The unit circle is a circle of radius 1 centred at the origin. For any angle θ measured from the positive x‑axis, the coordinates of the point where the terminal side meets the circle are (cosθ, sinθ). Therefore, cosθ = x, sinθ = y, and tanθ = y/x (x ≠ 0).

单位圆是一个以原点为圆心、半径为1的圆。对于从 x 轴正方向开始测量的任意角 θ，其终边与单位圆交点的坐标为 (cosθ, sinθ)。因此，cosθ = x，sinθ = y，tanθ = y/x (x ≠ 0)。

The sign of each trigonometric function depends on the quadrant in which the angle lies. A common memory aid is the ASTC diagram: All positive in Quadrant I, Sin positive in II, Tan positive in III, Cos positive in IV.

各三角函数的正负取决于角所在象限。常用口诀 ASTC：第一象限 All（全正），第二象限 Sin 正，第三象限 Tan 正，第四象限 Cos 正。

Using the unit circle, you can also deduce fundamental identities such as cos(−θ) = cosθ and sin(−θ) = −sinθ, revealing cosine as an even function and sine as an odd function.

通过单位圆，还可推得 cos(−θ) = cosθ，sin(−θ) = −sinθ，表明余弦是偶函数，正弦是奇函数。

3. Trigonometric Values of Special Angles | 特殊角的三角函数值

The exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and 90° (or their radian equivalents) can be derived from two special right‑angled triangles: the isosceles right triangle (45°−45°−90°) and half of an equilateral triangle (30°−60°−90°).

0°、30°、45°、60° 和 90°（或对应弧度）的正弦、余弦和正切的精确值可由两个特殊的直角三角形推导得出：等腰直角三角形（45°−45°−90°）以及等边三角形的一半（30°−60°−90°）。

The table below summarises these key values. You are expected to recall them without a calculator.

下表总结了这些关键数值。考试中要求脱离计算器直接记忆。

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

For angles beyond 90°, use symmetry and the unit circle to find exact values. For instance, sin(120°) = sin(60°) = √3/2 because sine is positive in the second quadrant.

对于大于 90° 的角，可利用对称性和单位圆求精确值。例如 sin(120°) = sin(60°) = √3/2，因为正弦在第二象限为正。

4. Graphs and Properties of Trigonometric Functions | 三角函数的图像与性质

The graph of y = sin x is a smooth wave that oscillates between −1 and 1, with a period of 2π. It crosses the x‑axis at integer multiples of π and reaches its maximum at π/2 + 2kπ, minimum at 3π/2 + 2kπ.

y = sin x 的图像是一条光滑的波浪线，在 -1 与 1 之间振荡，周期为 2π。它在 π 的整数倍处穿过 x 轴，在 π/2 + 2kπ 达到极大值，在 3π/2 + 2kπ 达到极小值。

The graph of y = cos x has the same amplitude and period, but it is a horizontal shift of the sine graph: cos x = sin(x + π/2). It starts at (0, 1) and completes one full cycle over 2π.

y = cos x 的图像具有相同的振幅和周期，但它是正弦图像的水平平移：cos x = sin(x + π/2)。它从 (0, 1) 开始，经过 2π 完成一个完整周期。

The graph of y = tan x has asymptotes at x = π/2 + kπ, a period of π, and passes through the origin. Its range is all real numbers, and it is strictly increasing between asymptotes.

y = tan x 的图像在 x = π/2 + kπ 处有渐近线，周期为 π，且过原点。其值域为全体实数，在相邻渐近线之间严格递增。

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

The two foundational identities are sin²θ + cos²θ = 1 and tanθ = sinθ / cosθ (cosθ ≠ 0). These can be derived directly from the unit circle definition where x² + y² = 1.

两个基本恒等式为 sin²θ + cos²θ = 1 和 tanθ = sinθ / cosθ (cosθ ≠ 0)。它们可直接由单位圆定义 x² + y² = 1 导出。

sin²θ + cos²θ = 1

The identity sin²θ + cos²θ = 1 allows you to express sin²θ in terms of cos²θ (and vice versa), which is extremely useful when simplifying expressions or proving other identities.

利用恒等式 sin²θ + cos²θ = 1 可将 sin²θ 用 cos²θ 表示（反之亦然），这在化简表达式或证明其他恒等式时极有帮助。

Another useful form is 1 + cot²θ = cosec²θ and 1 + tan²θ = sec²θ, obtained by dividing the fundamental identity by sin²θ or cos²θ. At AS level these are often introduced later but are worth noting.

另一个有用形式是 1 + cot²θ = cosec²θ 和 1 + tan²θ = sec²θ，可通过将基本恒等式除以 sin²θ 或 cos²θ 得到。在 AS 阶段这些常稍后引入，但值得留意。

6. Solving Simple Trigonometric Equations | 解简单三角方程

When solving an equation such as sin x = k, first find the principal value using a calculator or known exact values. Then use the symmetry of the sine graph or the unit circle to determine all solutions within the required interval.

解形如 sin x = k 的方程时，先用计算器或已知精确值求出主值，再利用正弦图像的对称性或单位圆求出在给定区间内的所有解。

For sin x = 0.5 on [0, 2π], the principal value is x = π/6. Since sine is also positive in the second quadrant, the second solution is x = π − π/6 = 5π/6.

对于 sin x = 0.5 在 [0, 2π] 上，主值为 x = π/6。由于正弦在第二象限也为正，第二个解为 x = π − π/6 = 5π/6。

For cos x = k, solutions are symmetric about the x‑axis on the unit circle: x = ± principal value + 2kπ. For tan x = k, solutions repeat every π: x = principal value + kπ.

对于 cos x = k，解关于单位圆 x 轴对称：x = ±主值 + 2kπ。对于 tan x = k，解每隔 π 重复一次：x = 主值 + kπ。

Always check the domain specified in the question. Express final answers in radians unless instructed otherwise, and list all values that satisfy the equation within the given range.

务必检查题目指定的定义域。除非另有说明，最终答案应用弧度表示，并列出在给定范围内满足方程的所有值。

7. The Sine Rule | 正弦定理

For any triangle ABC with side lengths a opposite A, b opposite B, and c opposite C, the sine rule states that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.

对于任意三角形 ABC，边 a 对角 A，边 b 对角 B，边 c 对角 C，正弦定理为 a/sin A = b/sin B = c/sin C = 2R，其中 R 为外接圆半径。

a/sin A = b/sin B = c/sin C

The sine rule is used when you know two angles and one side (AAS or ASA) to find a missing side, or when you know two sides and a non‑included angle (SSA). In the SSA case be aware of the ambiguous case: there may be two possible triangles.

正弦定理用于已知两角一边（AAS 或 ASA）求未知边，或已知两边及一个非夹角（SSA）的情形。在 SSA 情形下要留意模糊情况：可能存在两个不同的三角形。

For example, given A = 30°, B = 45°, and a = 10 cm, you can find b by setting 10/sin 30° = b/sin 45°, giving b = (10 sin 45°)/sin 30° = 10√2 cm.

例如，已知 A = 30°，B = 45°，a = 10 cm，可通过 10/sin 30° = b/sin 45° 求得 b = (10 sin 45°)/sin 30° = 10√2 cm。

8. The Cosine Rule | 余弦定理

The cosine rule links the three sides and one angle: a² = b² + c² − 2bc cos A. Equivalent versions exist for the other angles by cyclic permutation.

余弦定理联系三边与一角：a² = b² + c² − 2bc cos A。通过轮换可得到其他角的等价形式。

a² = b² + c² − 2bc cos A

This rule is applied when you have two sides and the included angle (SAS) to find the third side, or when you know all three sides (SSS) to find an angle via the rearranged form: cos A = (b² + c² − a²)/(2bc).

当已知两边及其夹角（SAS）求第三边，或已知三边（SSS）求角时，使用余弦定理。求角时可用变形公式：cos A = (b² + c² − a²)/(2bc)。

For instance, if b = 7, c = 8, and A = 60°, then a² = 7² + 8² − 2×7×8×cos 60° = 49 + 64 − 56 = 57, so a = √57. This avoids the ambiguity that can arise with the sine rule in some cases.

例如，若 b = 7，c = 8，A = 60°，则 a² = 7² + 8² − 2×7×8×cos 60° = 49 + 64 − 56 = 57，因此 a = √57。这避免了在某些情况下正弦定理可能产生的歧义。

9. Transformations of Trigonometric Graphs | 三角函数的图像变换

The general sinusoidal function can be written as y = A sin(B(x − C)) + D, where

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AS Mathematics: Trigonometric Functions – Key Concepts | AS 数学：三角函数 考点精讲