IB AQA Mathematics: Trigonometry – Key Points Explained | IB AQA 数学：三角函数 考点精讲

📚 IB AQA Mathematics: Trigonometry – Key Points Explained | IB AQA 数学：三角函数 考点精讲

Trigonometry forms the backbone of many topics in the IB AQA Mathematics syllabus, linking geometry, algebra, and calculus. Mastering trigonometric functions, identities, and equations is essential for success in both Paper 1 and Paper 2. This guide breaks down every key concept, from radian measure to inverse functions, with clear bilingual explanations and exam-focused insights.

三角函数是 IB AQA 数学课程中许多主题的支柱，连接了几何、代数和微积分。掌握三角函数、恒等式和方程对于 Paper 1 和 Paper 2 的考试成功至关重要。本指南从弧度制到反函数，逐项详解每个关键概念，提供清晰的双语解释和应试重点。

1. Radian Measure | 弧度制

s the radius. The relationship π rad = 180° allows conversion between degrees and radians.

弧度是当弧长等于半径时，圆心所对应的角。利用 π rad = 180° 的关系可以在角度和弧度之间进行转换。

values must be memorised: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

将角度转换为弧度，乘以 π/180；将弧度转换为角度，乘以 180/π。必须记住常见值：30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π。

In IB exams, always check the mode of your calculator. Many problems require exact values expressed in terms of π, so working in radian mode is crucial.

在 IB 考试中，务必检查计算器的模式。许多题目要求用 π 表示精确值，因此使用弧度模式至关重要。

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2. The Unit Circle | 单位圆

The unit circle is a circle of radius 1 centred at the origin. For any point P on the circle making an angle θ with the positive x-axis, the coordinates are (cos θ, sin θ).

单位圆是一个以原点为圆心、半径为 1 的圆。对于圆周上任意一点 P，与 x 轴正方向成 θ 角，其坐标为 (cos θ, sin θ)。

real numbers. It also shows that tan θ = sin θ / cos θ, which is the slope of the line OP.

这个几何定义将三角函数推广到所有实数。同时也可得出 tan θ = sin θ / cos θ，即直线 OP 的斜率。

Key angles on the unit circle yield well-known coordinate values. Use symmetry to deduce signs in each quadrant: All positive in Quadrant I, Sine positive in II, Tangent positive in III, Cosine positive in IV (ASTC).

单位圆上的特殊角度对应已知坐标值。利用对称性可推导各象限的符号：第一象限全正，第二象限正弦正，第三象限正切正，第四象限余弦正（ASTC口诀）。

Angle (θ)	0	π/6	π/4	π/3	π/2	π
sin θ	0	1/2	√2/2	√3/2	1	0
cos θ	1	√3/2	√2/2	1/2	0	−1
tan θ	0	1/√3	1	√3	undefined	0

Memorise this table; exact values are frequently tested, and you must be able to recall them without a calculator for special angles.

熟记上表；特殊角的精确值常出现在考试中，你必须能够不借助计算器直接写出。

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3. Sine, Cosine and Tangent | 正弦、余弦与正切

OH CAH TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

在直角三角形中，基本三角比由 SOH CAH TOA 定义：sin = 对边/斜边，cos = 邻边/斜边，tan = 对边/邻边。

These definitions are limited to acute angles, but the unit circle extends them to any angle. The sign of each function depends on the quadrant, as described by the ASTC diagram.

这些定义仅适用于锐角，但单位圆将其扩展到任意角。每个函数的符号取决于象限，如 ASTC 图所示。

= 1/cos θ, csc θ = 1/sin θ, and cot θ = 1/tan θ. They appear in identities and equations.

虽然 IB AQA 中较少见，但倒数函数 sec θ = 1/cos θ、csc θ = 1/sin θ 和 cot θ = 1/tan θ 会出现在恒等式和方程中。

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4. Trigonometric Graphs | 三角函数图像

at (0,0), with amplitude 1, period 2π, and zeros at multiples of π. It is symmetric about the origin (odd function).

y = sin x 的图像是一条从 (0,0) 出发的波形，振幅为 1，周期为 2π，在 π 的整数倍处为零。它关于原点对称（奇函数）。

in shape but shifted left by π/2. It starts at (0,1), has period 2π, and is symmetric about the y-axis (even function).

余弦图像 y = cos x 形状相同但向左平移了 π/2。它从 (0,1) 开始，周期为 2π，关于 y 轴对称（偶函数）。

x = π/2 + nπ. It passes through the origin and increases without bound between asymptotes.

正切图像 y = tan x 的周期为 π，在 x = π/2 + nπ 处有垂直渐近线。它经过原点，在渐近线之间无界递增。

You will often be asked to sketch these graphs or to identify key features: maximum, minimum, intercepts, and asymptotes. Understanding the parent graphs is essential before applying transformations.

考试中常要求画出这些图像或识别关键特征：最大值、最小值、截距和渐近线。在进行变换之前，必须理解原函数图像。

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5. Transformations of Trigonometric Functions | 三角函数的变换

The general sine function is written as y = A sin(B(x − C)) + D. Here, |A| is the amplitude, the period is 2π/|B|, C is the horizontal shift (phase shift), and D is the vertical shift.

一般正弦函数形式为 y = A sin(B(x − C)) + D。其中 |A| 是振幅，周期为 2π/|B|，C 是水平位移（相位移），D 是垂直位移。

y = A sin(Bx + C) + D → amplitude = |A|, period = 2π/|B|, phase shift = −C/B

类似地，若写成 y = A sin(Bx + C) + D，则振幅为 |A|，周期为 2π/|B|，相位移为 −C/B。

For cosine functions, the same rules apply. Be careful with factorisation: always write the argument as B(x − C) to identify the true horizontal shift.

对于余弦函数，同样规则适用。注意因式分解：始终将自变量写成 B(x − C) 的形式，以确定真实的水平位移。

Transformations greatly affect the graph's key points. For example, stretching vertically by a factor of 2 will double all y-coordinates, while a horizontal compression by 1/2 will halve the period.

变换会显著影响图像的关键点。例如，垂直拉伸为原来的 2 倍会使所有 y 坐标翻倍，而水平压缩为原来的 1/2 则会使周期减半。

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6. Trigonometric Identities | 三角恒等式

ty: sin² θ + cos² θ = 1. It is derived directly from the unit circle equation x² + y² = 1.

最基本的恒等式是毕达哥拉斯恒等式：sin² θ + cos² θ = 1。它直接来源于单位圆方程 x² + y² = 1。

sin² θ + cos² θ = 1

由此可推出：1 + tan² θ = sec² θ 和 1 + cot² θ = csc² θ。这些变形在化简表达式时经常用到。

= sec² θ and 1 + cot² θ = csc² θ. These are used extensively when simplifying expressions.

QA syllabus: sin 2θ = 2 sin θ cos θ, cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ.

IB AQA 大纲还要求掌握二倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ。

tan 2θ = 2 tan θ / (1 − tan² θ)

正切二倍角公式为 tan 2θ = 2 tan θ / (1 − tan² θ)。这些公式常用于求解方程和化简。

Proving identities and simplifying trigonometric expressions are common tasks. Always start with the more complex side, convert everything to sin and cos, and use the Pythagorean identity to simplify.

证明恒等式和化简三角表达式是常见题型。始终从较复杂的一边入手，将所有函数化为 sin 和 cos，并利用毕达哥拉斯恒等式进行化简。

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7. Solving Trigonometric Equations | 解三角方程

or calculator: x = π/6. Then use symmetry to find the second solution: x = π − π/6 = 5π/6.

求解如 sin x = ½ 在区间 0 ≤ x ≤ 2π 内的解，首先通过单位圆或计算器找到主解：x = π/6。然后利用对称性找到第二个解：x = π − π/6 = 5π/6。

sine, x = π/6 + 2πn or x = 5π/6 + 2πn. For cosine, x = ± θ + 2πn. For tangent, x = θ + πn.

通解包含周期项：正弦为 x = π/6 + 2πn 或 x = 5π/6 + 2πn；余弦为 x = ± θ + 2πn；正切为 x = θ + πn。

1, require solving for the inner angle first, adjusting the period, and then listing all solutions within the given interval.

对于函数内含有 x 倍数的情况，如 sin(2x) = 1，需先解内部角度，调整周期，然后列出给定区间内的所有解。

in x, solving the quadratic, and then solving the resulting basic trigonometric equations.

对于二次型方程 2 sin² x − sin x − 1 = 0，可通过设 u = sin x 化为二次方程求解，然后再解基本的三角方程。

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8. The Sine Rule and Cosine Rule | 正弦定理与余弦定理

ow either two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA).

对于任意三角形（不限于直角三角形），正弦定理为：a/sin A = b/sin B = c/sin C。当已知两角一边（AAS 或 ASA）或两边及其非夹角（SSA）时使用。

. It is applied when you know three sides (SSS) or two sides and the included angle (SAS).

余弦定理将边与其夹角联系起来：a² = b² + c² − 2bc cos A。当已知三边（SSS）或两边及其夹角（SAS）时使用。

Area = ½ ab sin C

三角形的面积也可通过三角函数求得：面积 = ½ ab sin C。务必记住，该角必须是两边之间的夹角。

a = ½ ab sin C. Always ensure the angle is the included angle between the two given sides.

在处理 SSA 情况时，请注意可能出现的“模糊情况”，即可能存在 0、1 或 2 个解，需要根据已知条件判断。

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9. Inverse Trigonometric Functions | 反三角函数

The inverse functions arcsin, arccos, and arctan undo sine, cosine, and tangent respectively. Their domains and ranges are restricted to make them true functions.

反三角函数 arcsin、arccos 和 arctan 分别撤销正弦、余弦和正切的运算。它们的定义域和值域受到限制，以确保其为真函数。

For arcsin x, the domain is [−1, 1] and the range is [−π/2, π/2]. For arccos x, domain [−1, 1], range [0, π]. For arctan x, domain is all real numbers, range (−π/2, π/2).

arcsin x 的定义域为 [−1, 1]，值域为 [−π/2, π/2]。arccos x 的定义域为 [−1, 1]，值域为 [0, π]。arctan x 的定义域为全体实数，值域为 (−π/2, π/2)。

In equation solving, applying an inverse function yields the principal value. You must then use graph knowledge to find all other solutions within the required interval.

在解方程时，运用反函数得到主值。然后必须利用图像知识找出所需区间内的所有其他解。

ad is the principal solution. The second solution in [0, 2π] is x = π − 0.775 ≈ 2.366 rad.

例如：若 sin x = 0.7，则 x = arcsin(0.7) ≈ 0.775 rad 是主解。在 [0, 2π] 区间内的第二个解为 x = π − 0.775 ≈ 2.366 rad。

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10. Applications and Exam Tips | 应用与考试技巧

Trigonometry models periodic phenomena such as tides, sound waves, and circular motion. IB questions may describe a real-world context and ask you to write a function, adjust parameters, or solve for a specific variable.

三角函数可用于模拟周期性现象，如潮汐、声波和圆周运动。IB 考题可能会描述实际情境，要求写出函数、调整参数或求解特定变量。

Always draw a diagram for geometry-based problems. Clearly label known sides, angles, and the unknown quantity. This reduces errors when applying the sine or cosine rule.

对于几何类题目，始终先画图。清晰标记已知边、角和未知量。这能减少运用正弦或余弦定理时的错误。

In exam conditions, manage your time by immediately setting your calculator to radian mode if needed. Write down exact values using π and surds unless told otherwise.

在考试中，如有需要，立即将计算器设为弧度模式以节省时间。除非题目另有要求，应使用 π 和根号写出精确值。

When solving trigonometric equations, always check that your solutions fall within the specified domain. Many marks are lost by providing extraneous or missing solutions.

解三角方程时，务必检查解是否落在指定区间内。许多失分都是由于提供了多余解或遗漏了解。

Practice proving identities by working from one side to the other, not by manipulating both simultaneously. Use substitution strategies for quadratic trigonometric equations and be comfortable factoring expressions like sin² x − sin x.

练习恒等式证明时，从一边推到另一边，而非同时操作两边。对于二次三角方程，熟练使用代换法，并能因式分解如 sin² x − sin x 这样的表达式。

Finally, familiarity with the exact value table and key identities will allow you to handle non-calculator sections confidently. Regular review of these fundamentals is the key to success.

最后，熟练掌握精确值表和关键恒等式，将使你在不允许使用计算器的部分也能自信作答。定期复习这些基础知识是成功的关键。

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