📚 IB Mathematics: Trigonometry Key Points | IB 数学：三角函数考点精讲
Trigonometry forms a fundamental pillar of the IB Mathematics curriculum, appearing across both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses at Standard and Higher Levels. Mastering trigonometric concepts—from radian measure to equations, identities, and calculus applications—is essential for success in Paper 1, Paper 2, and the internal assessment. This article condenses the key points you need to know, with clear explanations and practical examples that reflect typical IB exam style.
三角函数是 IB 数学课程的重要基石，无论是在分析与方法 (AA) 还是应用与解释 (AI) 的 SL 和 HL 课程中都频繁出现。掌握从弧度制到三角方程、恒等式以及微积分应用等核心概念，对在试卷一、试卷二和内部评估中取得好成绩至关重要。本文浓缩了必须掌握的考点，提供清晰的讲解和贴近 IB 考试风格的实例。
1. Angle Measurement: Degrees and Radians | 角度测量：度与弧度
Angles in IB Mathematics are commonly expressed in both degrees and radians. One radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. The conversion factor is π rad = 180 °, giving the formulas: radians = (π/180) × degrees and degrees = (180/π) × radians. You must be fluent in converting common angles such as 30°, 45°, 60°, 90°, 180°, and 360° into their radian equivalents (π/6, π/4, π/3, π/2, π, 2π).
IB 数学中角度通常以度和弧度两种方式表示。一弧度定义为弧长等于半径的圆弧所对的圆心角。换算关系为 π rad = 180 °，得出转换公式：弧度 = (π/180) × 度数，度数 = (180/π) × 弧度。必须熟练掌握常见角度 30°、45°、60°、90°、180° 和 360° 与弧度（π/6、π/4、π/3、π/2、π、2π）之间的转换。
When solving trigonometric equations or applying calculus to trigonometric functions, IB exam questions almost always assume the argument is in radians unless degrees are specified. Radians are essential for limit evaluations such as limx→0 (sin x)/x = 1, which only holds when x is in radians. Always check the mode of your calculator and the context of the problem.
在解三角方程或对三角函数进行微积分运算时，除非特别注明度数，IB 考题几乎都默认自变量采用弧度制。例如重要极限 limx→0 (sin x)/x = 1 仅在弧度制下成立。务必检查计算器的角度模式和题目上下文。
2. The Unit Circle and Trigonometric Definitions | 单位圆与三角定义
The unit circle, centred at the origin with radius 1, provides a visual and conceptual foundation for trigonometric functions. For any angle θ measured anticlockwise from the positive x‑axis, the coordinates of the point on the circle are (cos θ, sin θ). The tangent function is then defined as tan θ = sin θ / cos θ, which represents the slope of the line through the origin and that point.
单位圆以原点为圆心、半径为 1，为三角函数提供了直观的概念基础。对于从 x 轴正方向逆时针测量的任意角 θ，圆周上相应点的坐标为 (cos θ, sin θ)。正切函数定义为 tan θ = sin θ / cos θ，它表示过原点和该点的直线的斜率。
Using the unit circle, you can deduce the signs of sine, cosine, and tangent in each quadrant (ASTC: All, Sine, Tangent, Cosine positive in QI, QII, QIII, QIV respectively) and the exact values for key angles. For example, at θ = π/3, the point (1/2, √3/2) gives cos(π/3) = 1/2 and sin(π/3) = √3/2. Memorising the coordinates for 0, π/6, π/4, π/3, π/2, and their multiples helps quickly evaluate expressions without a calculator.
借助单位圆可以判断正弦、余弦和正切在各象限的符号（ASTC 法则：第一象限全正，第二象限仅正弦正，第三象限仅正切正，第四象限仅余弦正），并能求出特殊角的精确值。例如，当 θ = π/3 时，点的坐标为 (1/2, √3/2)，即 cos(π/3)=1/2，sin(π/3)=√3/2。熟记 0、π/6、π/4、π/3、π/2 及其整数倍对应的坐标，有助于在没有计算器的情况下快速求值。
The reciprocal trigonometric functions—cosecant (csc θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ = cos θ/sin θ)—also appear in certain identity and equation problems. They often require expressing everything in terms of sine and cosine first.
倒数三角函数——余割 csc θ = 1/sin θ、正割 sec θ = 1/cos θ 和余切 cot θ = 1/tan θ = cos θ/sin θ——也出现在一些恒等式和方程问题中。通常需要先将它们用正弦和余弦表示。
3. Graphs of Trigonometric Functions | 三角函数图像
The graph of y = sin x is a continuous wave that oscillates between –1 and 1, with period 2π and x‑intercepts at integer multiples of π. Its maximum value occurs at x = π/2 + 2kπ, minimum at x = 3π/2 + 2kπ. The graph is symmetric about the origin (odd function).
y = sin x 的图像是一条在 –1 与 1 之间连续波动的曲线，周期为 2π，在 π 的整数倍处与 x 轴相交。最大值出现在 x = π/2 + 2kπ，最小值在 x = 3π/2 + 2kπ。图像关于原点对称（奇函数）。
y = cos x is also a wave with amplitude 1 and period 2π, but it starts at (0,1) and is symmetric about the y‑axis (even function). Its x‑intercepts are at x = π/2 + kπ. The graph of y = tan x has vertical asymptotes at x = π/2 + kπ, period π, and passes through the origin. It is an odd function with no amplitude.
y = cos x 同样是振幅为 1、周期为 2π 的波形，但起点为 (0,1) 且关于 y 轴对称（偶函数），与 x 轴交于 x = π/2 + kπ。y = tan x 的图像在 x = π/2 + kπ 处有垂直渐近线，周期为 π，且过原点，为无振幅的奇函数。
Understanding the fundamental shapes allows you to apply transformations such as stretches, translations, and reflections. IB questions frequently ask you to sketch these graphs over a given domain or to use them to find the number of solutions to an equation like sin x = x/2.
理解基本形状后，就可以进行伸缩、平移和反射等图像变换。IB 考题常要求画出给定区间内的草图，或利用图像确定如 sin x = x/2 这类方程的解的个数。
4. Transformations of Trigonometric Graphs | 三角函数图像的变换
The general sine function can be 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 (principal axis y = d). For cosine and tangent, similar transformations apply, although tangent does not have an amplitude; instead, |a| affects the steepness of the curve.
一般正弦函数可写作 y = a sin(b(x – c)) + d。其中 |a| 为振幅，周期为 2π/|b|，c 为水平位移（相位移），d 为垂直位移（平衡轴 y = d）。余弦和正切函数也适用类似的变换，不过正切没有振幅，|a| 只影响曲线的陡峭程度。
When sketching transformed graphs, first mark the new principal axis, then the max/min points based on amplitude, and finally adjust the period and phase shift. Common IB questions include finding the equation of a graph from given features, or describing a sequence of transformations that maps one trigonometric curve onto another.
画变换后的图像时，先标出新的平衡轴，再根据振幅标出最大/最小值点，最后调整周期和相位移。常见的 IB 考题包括根据给定特征求图像方程，或描述将一条三角函数曲线映射为另一条的变换序列。
5. Fundamental Trigonometric Identities | 基本三角恒等式
The most essential identity is the Pythagorean identity: sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ yields 1 + cot²θ = csc²θ. These three identities form the basis for simplifying expressions and proving more complex identities.
最核心的恒等式是勾股恒等式：sin²θ + cos²θ = 1。两边同除以 cos²θ 得 1 + tan²θ = sec²θ，同除以 sin²θ 得 1 + cot²θ = csc²θ。这三个恒等式是化简表达式和证明更复杂恒等式的基础。
The reciprocal identities (csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ) and quotient identities (tan θ = sin θ/cos θ, cot θ = cos θ/sin θ) are equally important. IB strategies for proving identities: start from the more complex side, express everything in terms of sine and cosine, and look for common denominators or factorisations.
倒数恒等式（csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ）以及商恒等式（tan θ = sin θ/cos θ，cot θ = cos θ/sin θ）同样重要。IB 中证明恒等式的策略：从较复杂的一边入手，将所有函数用正弦和余弦表示，并寻找公分母或因式分解。
6. Compound Angle Formulas | 复合角公式
The compound angle formulas allow you to express the sine, cosine, and tangent of sums and differences of angles:
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).
复合角公式可以将两角和或差的三角函数展开：
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)。
These formulas are used to simplify expressions, solve equations like sin(x + 30°) = cos x, and to derive further identities such as 2 sin x cos y = sin(x + y) + sin(x – y). IB exams often ask for exact values of sin(75°) by writing it as sin(45° + 30°) and applying the formula. Remember to choose signs carefully according to the operation.
这些公式可用于化简表达式、解如 sin(x + 30°) = cos x 的方程，以及推导其他恒等式，例如 2 sin x cos y = sin(x + y) + sin(x – y)。IB 考试常要求将 sin(75°) 写成 sin(45°+30°) 并利用公式求其精确值。注意根据运算合理选择符号。
7. Double Angle Formulas | 二倍角公式
Setting A = B in the compound formulas gives the double angle identities:
sin 2A = 2 sin A cos A
cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A
tan 2A = 2 tan A / (1 – tan²A)
在复合角公式中令 A = B 即得二倍角公式：
sin 2A = 2 sin A cos A，
cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A，
tan 2A = 2 tan A / (1 – tan²A)。
The three forms of cos 2A are especially useful: the cos²A – sin²A form links to the Pythagorean identity, the 2 cos²A – 1 form helps with integration, and the 1 – 2 sin²A form is used to rewrite sin²A or cos²A as (1 – cos 2A)/2 or (1 + cos 2A)/2. These half‑angle rearrangements are vital for integrating squared trigonometric functions.
cos 2A 的三种形式各有用途：cos²A – sin²A 形式与勾股恒等式联系，2 cos²A – 1 形式便于积分，1 – 2 sin²A 形式可将 sin²A 或 cos²A 改写为 (1 – cos 2A)/2 或 (1 + cos 2A)/2。这些半角变形对积分平方三角函数至关重要。
8. Sum-to-Product and Product-to-Sum | 和差化积与积化和差
These identities are less frequently tested but still appear in challenging HL problems. Sum‑to‑product formulas include:
sin P + sin Q = 2 sin((P+Q)/2) cos((P–Q)/2),
sin P – sin Q = 2 cos((P+Q)/2) sin((P–Q)/2),
cos P + cos Q = 2 cos((P+Q)/2) cos((P–Q)/2),
cos P – cos Q = –2 sin((P+Q)/2) sin((P–Q)/2).
这些恒等式虽不常考，但仍可能出现在高难度的 HL 题目中。和差化积公式包括：
sin P + sin Q = 2 sin((P+Q)/2) cos((P–Q)/2)，
sin P – sin Q = 2 cos((P+Q)/2) sin((P–Q)/2)，
cos P + cos Q = 2 cos((P+Q)/2) cos((P–Q)/2)，
cos P – cos Q = –2 sin((P+Q)/2) sin((P–Q)/2)。
Product‑to‑sum formulas reverse these, expressing products like 2 sin A cos B as sin(A+B) + sin(A–B). They are useful for integrating products of sine and cosine, and for solving equations where a product equals a constant. Practice recognising which identity simplifies a given expression most rapidly.
积化和差公式是上述过程的逆用，例如 2 sin A cos B = sin(A+B) + sin(A–B)。这类公式对于积分正弦和余弦的乘积以及解某些乘积等于常数的方程很有帮助。应练习快速判断用哪个恒等式最有效。
9. Solving Trigonometric Equations | 解三角方程
Trigonometric equations often require finding all solutions within a specified interval, typically [0, 2π] or [–π, π]. Start by simplifying the equation using identities to obtain a basic form such as sin x = k, cos x = k, or tan x = k. For quadratics in sin x or cos x, substitute u = sin x (or cos x), solve for u, and then find the corresponding angles.
三角方程通常要求在指定区间（如 [0, 2π] 或 [–π, π]）内求出所有解。首先利用恒等式将方程化简为基本形式，如 sin x = k、cos x = k 或 tan x = k。对于关于 sin x 或 cos x 的二次方程，可设 u = sin x（或 cos x）求解 u，再求对应的角度。
Always consider the periodicity and symmetry of the trigonometric functions. For sin x = 0.5, the principal solution is π/6, but the general solutions are x = π/6 + 2kπ and x = 5π/6 + 2kπ. In IB, you must state all solutions in the given domain, often using the unit circle or graph to identify them. Carefully check if solutions need to be expressed as exact values or to a required decimal precision.
一定要考虑三角函数的周期性和对称性。例如 sin x = 0.5 的主解为 π/6，但通解为 x = π/6 + 2kπ 和 x = 5π/6 + 2kπ。在 IB 考试中，必须给出指定区间内的所有解，常借助单位圆或图像来识别。注意题目要求表达为精确值还是给定的小数精度。
For equations involving multiple angles such as sin 2x = √3/2, first solve for 2x, then divide by 2, and finally list all values of x that fall in the domain. Remember to double the interval when finding solutions for the multiple angle.
对于包含倍角的方程如 sin 2x = √3/2，先解 2x，再除以 2，最后列出所有落在给定区间内的 x 值。注意在求倍角的解时先要把区间对应地扩展。
10. Inverse Trigonometric Functions | 反三角函数
The inverse trigonometric functions arcsin x, arccos x, and arctan x return the angle whose sine, cosine, or tangent is x. Their domains and ranges are restricted to ensure they are functions:
arcsin: domain [–1, 1], range [–π/2, π/2];
arccos: domain [–1, 1], range [0, π];
arctan: domain ℝ, range (–π/2, π/2).
反三角函数 arcsin x、arccos x 和 arctan x 返回正弦、余弦或正切值为 x 的角度。为使其成为函数，定义域和值域被限制如下：
arcsin：定义域 [–1, 1]，值域 [–π/2, π/2]；
arccos：定义域 [–1, 1]，值域 [0, π]；
arctan：定义域 ℝ，值域 (–π/2, π/2)。
In IB HL, you may also need to differentiate inverse trigonometric functions: d/dx arcsin x = 1/√(1 – x²), d/dx arccos x = –1/√(1 – x²), d/dx arctan x = 1/(1 + x²). Composite functions such as arcsin(sin x) are not automatically equal to x; the output depends on whether the input lies in the principal range.
在 IB HL 课程中，还可能需要求反三角函数的导数：d/dx arcsin x = 1/√(1 – x²)，d/dx arccos x = –1/√(1 – x²)，d/dx arctan x = 1/(1 + x²)。复合函数如 arcsin(sin x) 并不自动等于 x，其值取决于输入是否在主值区间内。
Inverse trigonometric functions appear in calculus, vector geometry (finding angles between vectors), and in solving equations like cos x = 0.3 where the calculator gives a principal value, and you must find all solutions.
反三角函数出现在微积分、向量几何（求向量间夹角）以及解方程如 cos x = 0.3 中，此时计算器给出主值，需要自行求出所有解。
11. Sine and Cosine Rules | 正弦定理与余弦定理
For any non‑right‑angled triangle with sides a, b, c opposite angles A, B, C:
Sine Rule: a/sin A = b/sin B = c/sin C (= 2R, where R is the circumradius).
Cosine Rule: a² = b² + c² – 2bc cos A (similar for other sides).
对于任意非直角三角形，边 a、b、c 分别对应角 A、B、C：
正弦定理：a/sin A = b/sin B = c/sin C（= 2R，R 为外接圆半径）。
余弦定理：a² = b² + c² – 2bc cos A（其他边同理）。
The Sine Rule is used when given two angles and one side (AAS or ASA) or two sides and a non‑included angle (SSA, the ambiguous case). The ambiguous case can yield two possible triangles (acute and obtuse angles) when the known angle is acute and the opposite side length lies between the given side and its height. Always check the sum of angles to determine if one or two solutions exist.
当已知两角一边 (AAS 或 ASA) 或两边和其中一边的对角 (SSA，即模糊情况) 时，应用正弦定理。模糊情况中，当已知角为锐角且对边长度介于已知边及其高之间时，可能产生两个三角形（锐角和钝角）。应通过检查和是否为 180° 来判断解的数量。
The Cosine Rule is applied when given two sides and the included angle (SAS) or three sides (SSS). It is also useful to find an angle when all three sides are known. In IB, these rules are frequently linked with bearings, area, and 3D geometry problems.
当已知两边及其夹角 (SAS) 或三边 (SSS) 时，应用余弦定理。它也常用于已知三边求角度。IB 题目常将正余弦定理与方位角、面积和三维几何结合考查。
12. Area of a Triangle and Trigonometric Calculus | 三角形面积与三角微积分
The area of a triangle can be found using ½ ab sin C, where a and b are two sides and C is the included angle. This formula is especially powerful in problems where the perpendicular height is not directly given. Heron’s formula, Area = √(s(s–a)(s–b)(s–c)) with semiperimeter s = (a+b+c)/2, is another option for SSS cases.
三角形的面积可用公式 ½ ab sin C 计算，其中 a、b 为两边，C 为夹角。当垂直高度未直接给出时，该公式尤其有用。海伦公式（面积 = √(s(s–a)(s–b)(s–c))，其中半周长 s = (a+b+c)/2）是三边已知时的另一种选择。
In IB calculus, the derivatives of the basic trigonometric functions (with x in radians) are:
d/dx sin x = cos x,
d/dx cos x = –sin x,
d/dx tan x = sec² x.
Corresponding integrals are ∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C.
在 IB 微积分中，基本三角函数的导数（x 以弧度计）为：
d/dx sin x = cos x，
d/dx cos x = –sin x，
d/dx tan x = sec² x。
相应的积分为 ∫ sin x dx = –cos x + C，∫ cos x dx = sin x + C，∫ sec² x dx = tan x + C。
Chain rule extensions (e.g., d/dx sin(ax+b) = a cos(ax+b)) and reverse chain rule integrals are frequently examined. Trigonometric substitution, integration of products using identities like sin²x = ½(1–cos 2x), and volumes of revolution involving trigonometric curves also appear, especially in HL papers.
链式法则的扩展（如 d/dx sin(ax+b) = a cos(ax+b)）以及反向链式法则的积分题常出现在考试中。三角替换、利用恒等式如 sin²x = ½(1–cos 2x) 对乘积进行积分，以及涉及三角函数曲线的旋转体积，尤其常见于 HL 试卷。
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