A-Level CIE Maths: Trigonometry Key Points Revision | A-Level CIE 数学：三角函数考点精讲

📚 A-Level CIE Maths: Trigonometry Key Points Revision | A-Level CIE 数学：三角函数考点精讲

Trigonometry is a core topic in CIE A-Level Mathematics, appearing in both Pure Mathematics 1 & 3. Mastering trigonometric concepts is essential for solving equations, modelling periodic phenomena, and later calculus applications. This article summarises all the key points you need to know for the exam.

三角学是 CIE A-Level 数学的核心主题，出现在纯数学1和3中。掌握三角函数概念对于解方程、建立周期现象模型以及后续的微积分应用至关重要。本文总结了你需要掌握的考试关键点。

1. Radian Measure | 弧度制

Radians are an alternative to degrees for measuring angles. By definition, π radians = 180°. To convert, multiply degrees by π/180, or radians by 180/π. The arc length formula s = rθ and sector area A = ½ r²θ only hold when θ is in radians. Understanding radian measure is crucial for calculus and trigonometric limits.

弧度是角度的另一种度量单位。定义上，π 弧度 = 180°。转换时，角度乘以 π/180，或弧度乘以 180/π。弧长公式 s = rθ 和扇形面积 A = ½ r²θ 仅在 θ 以弧度为单位时成立。理解弧度制对微积分和三角极限至关重要。

2. Trigonometric Ratios & The Unit Circle | 三角函数比与单位圆

The unit circle defines sine and cosine for any angle: cos θ is the x‑coordinate, sin θ is the y‑coordinate of a point on the circle. tan θ = sin θ / cos θ. Exact values for 0, π/6, π/4, π/3, π/2 etc. must be known. The signs in each quadrant can be remembered by ‘All Students Take Calculus’.

单位圆定义了任意角的正弦和余弦：cos θ 是圆上点的 x 坐标，sin θ 是 y 坐标。tan θ = sin θ / cos θ。必须牢记 0, π/6, π/4, π/3, π/2 等精确值。各象限的符号规则可用 ‘All Students Take Calculus’ 记忆。

3. Graphs of Trigonometric Functions | 三角函数图像

Sketch y = sin x, y = cos x, y = tan x. sin x and cos x have period 2π, amplitude 1. tan x has period π and vertical asymptotes at x = π/2 + kπ. Note symmetries: sin is odd, cos is even. Understanding these basic shapes is vital before applying transformations.

掌握 y = sin x, y = cos x, y = tan x 的图像。sin x 和 cos x 的周期为 2π，振幅为 1。tan x 的周期为 π，并在 x = π/2 + kπ 处有垂直渐近线。注意对称性：sin 是奇函数，cos 是偶函数。在应用变换前理解这些基本形状至关重要。

4. Transformations of Trig Graphs | 三角图像变换

For y = a sin(bx + c) + d: amplitude = |a|, period = 2π/|b|, phase shift = −c/b, vertical shift = d. Analogous rules apply to cos and tan. Exam questions often ask you to find parameters from a given graph or describe a sequence of transformations.

对于 y = a sin(bx + c) + d：振幅 = |a|，周期 = 2π/|b|，相位平移 = −c/b，垂直平移 = d。cos 和 tan 类似。考题常要求根据给定图像求参数，或描述一系列变换。

5. Trigonometric Identities | 三角恒等式

Core identities: tan θ ≡ sin θ / cos θ; sin² θ + cos² θ ≡ 1. From these we derive 1 + tan² θ ≡ sec² θ and 1 + cot² θ ≡ csc² θ. Use them to simplify expressions, prove identities, and solve equations. Be comfortable with algebraic manipulation and recognising hidden forms.

核心恒等式：tan θ ≡ sin θ / cos θ；sin² θ + cos² θ ≡ 1。由此可推导出 1 + tan² θ ≡ sec² θ 和 1 + cot² θ ≡ csc² θ。用它们来化简表达式、证明恒等式和解方程。熟练进行代数操作并识别隐藏形式。

6. Compound Angle Formulas | 和角公式

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 deriving double‑angle formulas, solving equations, and integration.

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)
这些公式是推导倍角公式、解方程和积分的基础。

7. Double Angle Formulas | 倍角公式

sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A
tan 2A = 2 tan A / (1 − tan² A)
Choosing the right form of cos 2A is particularly helpful when integrating powers of sine and cosine.

sin 2A = 2 sin A cos A
cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A
tan 2A = 2 tan A / (1 − tan² A)
在积分正弦和余弦的幂时，选择合适的 cos 2A 形式尤为有用。

8. Solving Trigonometric Equations | 解三角方程

First use identities to rewrite the equation in terms of a single trig function. Find the principal value using an inverse function or known exact values. Then determine all solutions within the given interval using the CAST diagram or graphs. Always check for extraneous solutions introduced by squaring, and verify that answers lie in the required range.

首先利用恒等式将方程化为只含一个三角函数的式子。用反函数或已知精确值求出主值。然后用 CAST 图或图像找出给定区间内的所有解。始终检查平方产生的增根，并验证答案在所需范围内。

9. Expressing a sin x + b cos x | 辅助角公式

Write a sin x + b cos x as R sin(x ± α) or R cos(x ± α), where R = √(a² + b²) and tan α = b/a (adjust for the chosen form). This transformation is extremely useful for finding maximum/minimum values and solving equations of the form a sin x + b cos x = c.

将 a sin x + b cos x 表示为 R sin(x ± α) 或 R cos(x ± α)，其中 R = √(a² + b²)，tan α = b/a（根据所选形式调整）。此变换在求最大值/最小值和解形如 a sin x + b cos x = c 的方程时非常有用。

10. Inverse Trigonometric Functions (P3) | 反三角函数

y = arcsin x, y = arccos x, y = arctan x. Their domains and ranges are: arcsin x: [−1, 1] → [−π/2, π/2]; arccos x: [−1, 1] → [0, π]; arctan x: ℝ → (−π/2, π/2). The derivatives are d/dx (arcsin x) = 1/√(1−x²), d/dx (arccos x) = −1/√(1−x²), d/dx (arctan x) = 1/(1+x²). These appear in calculus problems and substitution integrals.

y = arcsin x, y = arccos x, y = arctan x。它们的定义域和值域为：arcsin x: [−1, 1] → [−π/2, π/2]；arccos x: [−1, 1] → [0, π]；arctan x: ℝ → (−π/2, π/2)。导数为 d/dx (arcsin x) = 1/√(1−x²)，d/dx (arccos x) = −1/√(1−x²)，d/dx (arctan x) = 1/(1+x²)。这些会出现在微积分和换元积分中。

11. Derivatives of Trigonometric Functions (P3) | 三角函数的导数

Standard derivatives: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x. Extend with the chain rule for sin(kx), cos(ax + b), etc. You may also need derivatives of sec x, csc x, cot x. Combined with product, quotient and chain rules, trig differentiation underpins many optimisation and rates of change problems.

标准导数：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x。利用链式法则扩展到 sin(kx), cos(ax + b) 等。你可能还需要 sec x, csc x, cot x 的导数。结合乘积法则、商法则和链式法则，三角微分是许多优化和变化率问题的基础。

12. Integrals of Trigonometric Functions (P3) | 三角函数的积分

∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C. For ∫ sin(ax + b) dx use the reverse chain rule, dividing by a. To integrate sin² x or cos² x, rewrite using double‑angle formulas. Substitution and integration by parts frequently involve trig functions. Always include the constant of integration C.

∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C。对于 ∫ sin(ax + b) dx，使用反向链式法则并除以 a。积分 sin² x 或 cos² x 时，用倍角公式改写。换元积分和分部积分常涉及三角函数。别忘了积分常数 C。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level CIE Maths: Trigonometry Key Points Revision | A-Level CIE 数学：三角函数 考点精讲