📚 IB Mathematics: Trigonometric Functions and Their Core Properties | IB数学：三角函数及其核心性质
Trigonometric functions form a cornerstone of the IB Mathematics curriculum, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. From modelling periodic phenomena to solving complex equations, a thorough command of trigonometry is essential for top marks in both Paper 1 and Paper 2.
三角函数是IB数学课程的核心基石，在分析与方法（AA）和应用与解释（AI）两门课程中均占据重要地位。从模拟周期现象到求解复杂方程，熟练掌握三角函数是取得高分的关键。
1. The Unit Circle and Radian Measure | 单位圆与弧度制
ert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π.
弧度制是所有高等三角学的基础。一弧度是弧长等于半径时所对应的圆心角。整圆为2π弧度，由此得到关键换算：180° = π 弧度。将度数化为弧度需要乘以 π/180，将弧度化为度数则乘以 180/π。
secant and cotangent are defined as csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ.
单位圆（半径为1）为任意角θ定义六个三角函数。对于圆上一点P(x, y)，有 sin θ = y，cos θ = x，tan θ = y/x（x ≠ 0）。余割、正割和余切等倒数函数定义为 csc θ = 1/sin θ，sec θ = 1/cos θ，cot θ = 1/tan θ。
sin² θ + cos² θ = 1, tan θ = sin θ / cos θ
The quadrant signs are vital. The mnemonic "All Students Take Calculus" (ASTC) helps: all six functions are positive in Quadrant I, sine is positive in Quadrant II, tangent is positive in Quadrant III, and cosine is positive in Quadrant IV. Be equally careful with the signs of the reciprocal functions.
象限符号至关重要。助记口诀"All Students Take Calculus"（ASTC法）可辅助记忆：第一象限所有函数为正，第二象限正弦为正，第三象限正切为正，第四象限余弦为正。对倒数函数的符号也要同样小心。
The exact values for special angles must be memorised. They appear constantly in IB papers, both in algebraic manipulation and in geometric problems.
特殊角的精确值必须牢记。它们在IB试卷中频繁出现，既用于代数运算，也用于几何问题。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
2. Graphs of Trigonometric Functions | 三角函数的图像
are periodic with period 2π and range [-1, 1]. The graph of y = tan x has period π, range ℝ rigin and reaches its maximum at x = π/2; the cosine graph starts at its maximum at x = 0.
y = sin x 与 y = cos x 的图像周期为2π，值域为[-1, 1]。y = tan x 的图像周期为π，值域为全体实数，在 x = π/2 + kπ（k为任意整数）处有垂直渐近线。正弦图像过原点，在 x = π/2 处达到最大值；余弦图像在 x = 0 处从最大值开始。
Key features to identify on a trig graph are: amplitude (vertical stretch from the midline), period (horizontal length of one full cycle), phase shift (horizontal translation) and vertical displacement (the midline). Understanding how these features correspond to the parameters in the equation is non-negotiable.
识别三角函数图像的关键特征包括：振幅（相对中线的纵向拉伸）、周期（一个完整循环的水平长度）、相移（水平平移）以及垂直位移（中线位置）。理解这些特征与方程参数的对应关系是必备能力。
In IB exams, you are often asked to read these parameters directly from a graph or to write the equation of a given curve. Practise matching equations to graphs quickly, as this appears frequently in Paper 1, where no calculator is allowed.
IB考试中，常要求你直接从图像读取这些参数，或写出给定曲线的方程。练习快速将方程与图像对应，因为这类题目在不可使用计算器的卷一中频繁出现。
3. Fundamental Identities | 基本恒等式
two derived identities are obtained by dividing through by cos² θ and sin² θ respectively:
毕达哥拉斯恒等式是三角函数化简的基石。除 sin² θ + cos² θ = 1 外，分别除以 cos² θ 和 sin² θ 可得到另外两个衍生恒等式：
tan² θ + 1 = sec² θ, 1 + cot² θ = csc² θ
is to convert everything to sine and cosine first, then simplify using algebraic manipulation.
这些恒等式用于化简表达式、证明其他恒等式以及解方程。IB标准题型是要求证明一个三角恒等式，例如 (1 − cos² x) / sin x = sin x。最稳妥的方法是将所有项先转换为正弦和余弦，再通过代数运算化简。
When proving identities, work with one side only and never move terms across the equals sign. Show every step clearly — marks are awarded for method as well as the final result. A common error is treating an identity as an equation and applying the same operation to both sides; avoid this.
证明恒等式时，只对一边进行变形，切勿将项移到等号另一边。每一步都要清晰写出——方法过程与最终结果同样得分。常见错误是把恒等式当作方程，对两边同时进行运算；务必避免。
4. Compound Angle Formulas | 和角公式
s of the trigonometric functions of A and B. These are provided in the IB formula booklet, but you must know when and how to apply them confidently.
和角公式将 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 表示为 A 和 B 的三角函数形式。这些公式在IB公式手册中提供，但你必须熟练且恰当地应用它们。
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)
as a single sine or cosine, which helps solve equations of the form a sin x + b cos x = c.
一个经典应用是利用已知精确值求75° = 45° + 30° 或15° = 45° − 30° 等角的精确值。另一个重要用途是逆用公式：将三角表达式的乘积识别为单个正弦或余弦，这有助于求解形如 a sin x + b cos x = c 的方程。
The compound angle formulas also underpin the derivation of the double angle formulas in the next section, so make sure you understand the structure rather than merely memorising outputs.
和角公式也是下一节倍角公式的推导基础，因此要理解其结构而非仅仅记住结果。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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