📚 A-Level WJEC Maths Trigonometry: Key Points | A-Level WJEC 数学：三角函数考点精讲
This revision guide covers everything you need to master the WJEC A-Level Mathematics trigonometry topics, from basic ratios and exact values to solving equations, compound angles, and the harmonic form. Each section presents key facts and worked-style explanations in both English and Chinese, designed to build confidence for your exams.
本复习指南涵盖了攻克 WJEC A-Level 数学三角函数部分所需的所有内容，从基本比和精确值到解方程、和角公式以及辅助角形式。每个小节都以中英双语呈现关键概念和例题式讲解，旨在为你的考试建立信心。
1. Trigonometric Ratios and Exact Values | 三角比与精确值
The three primary trigonometric ratios for a right-angled triangle are defined by ‘SOH CAH TOA’: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent. You must memorise the exact values for 0°, 30°, 45°, 60° and 90°, which also correspond to the radian measures 0, π/6, π/4, π/3 and π/2.
直角三角形的三个基本三角比由 “SOH CAH TOA” 定义：正弦 = 对边/斜边，余弦 = 邻边/斜边，正切 = 对边/邻边。你必须熟记 0°, 30°, 45°, 60° 和 90° 的精确值，它们也对应弧度制 0, π/6, π/4, π/3 和 π/2。
| θ (degrees) | θ (radians) | sinθ | cosθ | tanθ |
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Beyond the first quadrant, the sign of each ratio depends on the quadrant. Use the ‘CAST’ diagram to remember which functions are positive in which quadrant.
第一象限之外，每个比的符号取决于象限。利用“CAST”图来记住各函数在哪个象限为正。
2. Radian Measure and Circular Arcs | 弧度制与圆弧
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Conversion between degrees and radians is given by 180° = π rad. Therefore, to convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.
一弧度是圆心角所对的弧长等于半径时所对应的角度。度与弧度之间的转换为 180° = π 弧度。因此，度转换为弧度乘以 π/180；弧度转换为度乘以 180/π。
Arc length s = rθ
Area of sector A = ½r²θ
These formulas hold only when θ is measured in radians. For a segment, subtract the area of the triangle from the sector area.
这两个公式仅在 θ 以弧度为单位时成立。对于弓形面积，用扇形面积减去三角形面积即可。
3. Graphs of Sine, Cosine and Tangent | 正弦、余弦、正切图形
The graph of y = sin x is a wave with amplitude 1 and period 2π. It crosses the x-axis at multiples of π and reaches maxima at π/2 + 2kπ and minima at 3π/2 + 2kπ. The graph of y = cos x has the same shape but starts at (0,1) and is shifted left by π/2.
y = sin x 的图形是一个振幅为 1、周期为 2π 的波形，在 π 的整数倍处与 x 轴相交，在 π/2 + 2kπ 处取得最大值，在 3π/2 + 2kπ 处取得最小值。y = cos x 的图形形状相同，但从 (0,1) 开始，且向左平移了 π/2。
The tangent function y = tan x has period π and vertical asymptotes at x = π/2 + kπ. Its range is all real numbers, and the graph repeats every π units. There is no amplitude for the tangent curve.
正切函数 y = tan x 的周期为 π，且在 x = π/2 + kπ 处有垂直渐近线。其值域为全体实数，图形每 π 个单位重复一次。正切曲线没有振幅。
4. Transforming Trigonometric Graphs | 三角图形变换
Transformations of trigonometric graphs follow the general form y = a sin(bx + c) + d or the equivalent with cos/tan. |a| gives the amplitude (vertical stretch), the period is 2π/|b| (for sin and cos, or π/|b| for tan), c causes a horizontal shift of –c/b, and d gives a vertical translation.
三角函数的图形变换遵循一般形式 y = a sin(bx + c) + d（余弦和正切同理）。|a| 给出振幅（纵向伸缩），周期为 2π/|b|（正弦和余弦）或 π/|b|（正切），c 引起 –c/b 的水平平移，d 给出纵向平移。
Always factor the argument first, e.g., y = sin(2x + π/3) = sin[2(x + π/6)], to see the correct phase shift. Mastering these translations lets you sketch any trigonometric graph quickly.
始终先对自变量进行因式分解，例如 y = sin(2x + π/3) = sin[2(x + π/6)]，这样才能得到正确的相位移动。掌握这些平移将使你快速画出任何三角函数图形。
5. Basic Trigonometric Identities | 基本三角恒等式
The foundational identity linking sine and cosine is the Pythagorean identity:
联系正弦与余弦的基本恒等式是毕达哥拉斯恒等式：
sin²θ + cos²θ ≡ 1
Dividing through by cos²θ gives tan²θ + 1 ≡ sec²θ, and dividing by sin²θ gives 1 + cot²θ ≡ cosec²θ. However, for WJEC pure maths, the most frequently used forms are the first identity together with tanθ ≡ sinθ / cosθ.
两边同除以 cos²θ 得到 tan²θ + 1 ≡ sec²θ，同除以 sin²θ 得到 1 + cot²θ ≡ cosec²θ。但对 WJEC 纯数而言，最常用的是第一个恒等式以及 tanθ ≡ sinθ / cosθ。
You will also need to use these identities to simplify expressions and to prove other relationships – always start from the more complicated side and work towards the simpler side.
你还需要使用这些恒等式对表达式进行化简，并证明其他关系——始终从较复杂的一边入手，向较简单的一边变形。
6. Compound Angle Formulae | 和角公式
The compound angle (addition) formulae allow you to find the sine, cosine and tangent of sums or differences of angles. They are given in your formula booklet, but memorising them speeds up problem-solving.
和角公式可用于求两角和或差的正弦、余弦和正切。公式手册会提供，但熟记它们能加快解题速度。
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)
These are essential for expanding expressions like sin 75° = sin(45°+30°) and for simplifying integrals in later topics. Watch the signs carefully when the angles differ.
这些公式对展开类似 sin 75° = sin(45°+30°) 的表达式以及后续积分化简至关重要。当角度相减时要特别注意符号。
7. Double Angle Formulae | 倍角公式
The double angle formulae are special cases of the compound angle formulae. The three forms for cosine are equally important in solving equations.
倍角公式是和角公式的特例。余弦的三种形式在解方程时同等重要。
sin 2A ≡ 2 sin A cos A
cos 2A ≡ cos²A – sin²A ≡ 2 cos²A – 1 ≡ 1 – 2 sin²A
tan 2A ≡ 2 tan A / (1 – tan²A)
Rearranging cos 2A helps express sin²A and cos²A in terms of cos 2A, which is useful for integrating powers of trig functions and for solving trigonometric equations.
重新整理 cos 2A 可将 sin²A 和 cos²A 用 cos 2A 表示，这对于积分三角函数的幂次以及解三角方程都很有用。
8. Harmonic Form R sin(θ ± α) | 辅助角公式 R sin(θ ± α)
Any expression of the type a sinθ ± b cosθ can be rewritten as R sin(θ ± α) or R cos(θ ± α), where R = √(a² + b²) and α is an acute angle such that tan α = b/a (depending on the chosen form).
任何形如 a sinθ ± b cosθ 的表达式都可以重写为 R sin(θ ± α) 或 R cos(θ ± α)，其中 R = √(a² + b²)，且 α 是一个锐角满足 tan α = b/a（取决于所选形式）。
a sinθ + b cosθ ≡ R sin(θ + α), with R cos α = a, R sin α = b
a sinθ – b cosθ ≡ R sin(θ – α), with R cos α = a, R sin α = b
This wave-form conversion is particularly powerful for finding the maximum and minimum values of an expression and for solving equations where the angle terms are mixed.
这种波形转换在求表达式的最大值和最小值，以及求解包含混合三角项的方程时尤为强大。
9. Solving Trigonometric Equations | 解三角方程
Solving trigonometric equations in a given interval requires you to consider all angles that satisfy the basic equation. First, find the principal value using inverse functions, then generate additional solutions using the periodic properties and the CAST diagram.
在给定区间内解三角方程需要你找出满足基本方程的所有角。首先用反函数求出主值，然后利用周期性和 CAST 图生成其他解。
For example, solve sin x = 0.5 for 0 ≤ x ≤ 2π. The principal value is π/6, and symmtery gives π – π/6 = 5π/6. Always check your solutions lie within the specified interval and adjust for multiples of the period if necessary.
例如，解 sin x = 0.5 在 0 ≤ x ≤ 2π 内。主值为 π/6，通过对称性得到 π – π/6 = 5π/6。务必检查所有解都在指定区间内，必要时加上周期的整数倍。
More complex equations may involve identities to reduce to a single trig function. For sin 2x = cos x, use the double angle formula to get 2 sin x cos x = cos x, then factorise and solve separately.
更复杂的方程可能需要动用恒等式化归为单一三角函数。对 sin 2x = cos x，使用倍角公式得 2 sin x cos x = cos x，然后因式分解并分别求解。
10. Sine and Cosine Rules & Triangle Area | 正弦和余弦定理与三角形面积
For non-right-angled triangles, the sine rule and cosine rule are essential tools. Use the sine rule when you know two angles and a side (AAS) or two sides and a non-included angle (SSA); use the cosine rule for SAS or SSS.
对非直角三角形，正弦定理和余弦定理是必不可少的工具。当已知两角一边（AAS）或两边及一个非夹角（SSA）时使用正弦定理；对于已知两边及夹角（SAS）或三边（SSS）时使用余弦定理。
Sine rule: a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² – 2bc cos A
Area = ½ab sin C
Remember that the sine rule may give an ambiguous case (two possible triangles) when you know two sides and a non-included acute angle. Always test whether the supplement (180° – angle) yields a valid second triangle.
注意，当你已知两边和一个非夹锐角时，正弦定理可能产生两解情况（两个可能的三角形）。务必检验补角（180° – 角）是否能构成有效的第二个三角形。
11. Inverse Trigonometric Functions | 反三角函数
The inverse functions arcsin x, arccos x and arctan x are the reflections of the restricted portions of the sine, cosine and tangent graphs in the line y = x. Their domains and ranges are specifically restricted so that each is a proper function.
反三角函数 arcsin x, arccos x 和 arctan x 分别是正弦、余弦和正切图形限制部分关于直线 y = x 的反射。它们的定义域和值域被特别限制以保证每个反函数都满足函数定义。
| arcsin x | Domain: –1 ≤ x ≤ 1 | Range: –π/2 ≤ y ≤ π/2 |
| arccos x | Domain: –1 ≤ x ≤ 1 | Range: 0 ≤ y ≤ π |
| arctan x | Domain: x ∈ ℝ | Range: –π/2 < y < π/2 |
The values arcsin(–x) = –arcsin x and arctan(–x) = –arctan x show odd symmetry, while arccos(–x) = π – arccos x. These identities help when solving equations that yield negative arguments.
arcsin(–x) = –arcsin x 和 arctan(–x) = –arctan x 表现出奇对称性，而 arccos(–x) = π – arccos x。这些恒等式在处理带负值参数的方程时很有帮助。
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