📚 Trigonometric Functions and Their Properties | 三角函数及其性质
Trigonometric functions form a cornerstone of IB Mathematics, appearing across both Analysis and Approaches and Applications and Interpretation. They describe periodic behaviour found in waves, oscillations, and circular motion. Understanding their definitions, graphs, and key properties such as period, amplitude, phase shift, and symmetry is essential for solving equations and modelling real-world scenarios. This article systematically reviews these concepts, from radian measure to identities, ensuring a solid foundation for exam success.
三角函数是IB数学的基石，贯穿于分析与方法、应用与解释两门课程。它们描述波动、振荡和圆周运动中的周期性行为。理解其定义、图形及周期、振幅、相位移和对称性等关键性质，对于解方程和建立现实世界模型至关重要。本文系统回顾从弧度制到恒等式的这些概念，为考试成功打下坚实基础。
1. Angle Measure and Radians | 角度度量与弧度制
subtended by an arc of length equal to the radius. The conversion factor is π rad = 180°.
角度可以用度数度量，但在微积分和三角分析中，弧度是标准。一弧度是弧长等于半径时所对的圆心角。换算关系为 π 弧度 = 180°。
an measure allows direct use of arc length formula s = rθ and simplifies derivative rules.
关键关系：90° = π/2 弧度，60° = π/3 弧度，45° = π/4 弧度，30° = π/6 弧度。弧度制使得弧长公式 s = rθ 可以直接使用，并简化了求导法则。
In IB, students must be comfortable switching between degrees and radians and setting calculators accordingly. All graphs of trigonometric functions are drawn with the x-axis in radians unless otherwise stated.
在IB中，学生必须熟练地在角度与弧度之间切换，并相应地设置计算器。除非另有说明，所有三角函数图像的 x 轴都以弧度为单位。
2. The Unit Circle and Trigonometric Definitions | 单位圆与三角函数定义
The unit circle (radius 1, centre (0,0)) provides a powerful geometric interpretation. For a point P on the circle at angle θ from the positive x-axis, the coordinates are (cos θ, sin θ). The tangent is defined as sin θ / cos θ, which corresponds to the length of the tangent segment when cos θ ≠ 0.
单位圆（半径为1，圆心在原点）提供了一种强大的几何解释。对于圆上与 x 轴正方向成角 θ 的一点 P，其坐标为 (cos θ, sin θ)。正切定义为 sin θ / cos θ，并且当 cos θ ≠ 0 时对应于切线段的长度。
The definitions extend to all real angles, not just acute ones. Cosine and sine are the x- and y-coordinates, so their values repeat every full rotation of 2π radians. The signs in each quadrant follow the CAST rule.
这些定义扩展到所有实数角，而不仅是锐角。余弦和正弦分别是 x 坐标和 y 坐标，因此其值每转完整一圈（2π 弧度）重复一次。各个象限的符号遵循 CAST 规则。
Domain of sin and cos is all real numbers; range is [–1, 1]. For tan θ, domain excludes θ = π/2 + nπ (vertical asymptotes), range is all reals.
正弦和余弦的定义域为全体实数，值域为 [–1, 1]。正切函数的定义域不包括 θ = π/2 + nπ（垂直渐近线），值域为全体实数。
3. The Sine and Cosine Functions | 正弦与余弦函数
The sine function f(x) = sin x starts at 0, rises to 1 at π/2, returns to 0 at π, drops to –1 at 3π/2, and completes a cycle at 2π. Its shape is a smooth wave.
正弦函数 f(x) = sin x 从 0 开始，在 π/2 处升至 1，在 π 处回到 0，在 3π/2 处降至 –1，并于 2π 完成一个周期。其形状是一条光滑的波形。
The cosine function f(x) = cos x begins at 1, falls to 0 at π/2, reaches – urns to 0 at 3π/2, and back to 1 at 2π. It is essentially a sine wave shifted left by π/2.
余弦函数 f(x) = cos x 从 1 开始，在 π/2 处降至 0，在 π 处达到 –1，于 3π/2 回到 0，并在 2π 处返回 1。它本质上是一条向左平移 π/2 的正弦波。
iable everywhere. The relationship cos x = sin(x + π/2) highlights their phase connection.
两者的周期均为 2π，振幅为 1，处处连续且可导。关系式 cos x = sin(x + π/2) 突出了它们之间的相位联系。
4. The Tangent Function | 正切函数
n, has vertical asymptotes at x = π/2 + nπ, and repeats every π units, so its period is π.
正切函数 f(x) = tan x = sin x / cos x 具有完全不同的形状。它穿过原点，在 x = π/2 + nπ 处有垂直渐近线，并且每 π 个单位重复一次，因此其周期为 π。
Between asymptotes, the curve increases strictly from –∞ to +∞. Its range is all real numbers, and it is odd: tan(–x) = –tan x. The graph is symmetric about the origin.
在渐近线之间，曲线从 –∞ 严格递增到 +∞。其值域为全体实数，且是奇函数：tan(–x) = –tan x。图像关于原点对称。
Understanding asymptotes and the unbounded nature is critical when solving equations or evaluating limits.
理解渐近线和无界性质对于解方程或计算极限至关重要。
5. Reciprocal Trigonometric Functions | 倒数三角函数
s x / sin x. They appear less frequently but are important for identities and derivatives.
定义了三个倒数函数：正割 sec x = 1/cos x，余割 csc x = 1/sin x，余切 cot x = 1/tan x = cos x / sin x。它们出现频率较低，但对于恒等式和求导很重要。
d when cos x = 0, i.e., x = π/2 + nπ. Graphs show branches with asymptotes and ranges of (–∞, –1] ∪ [1, ∞) for sec and csc.
它们的定义域不包括分母为零的值。例如，当 cos x = 0，即 x = π/2 + nπ 时，sec x 无定义。图像呈现分支，有渐近线，且 sec 和 csc 的值域为 (–∞, –1] ∪ [1, ∞)。
Cot x has period π, vertical asymptotes where sin x = 0, and decreases from +∞ to –∞ between them. Its graph resembles tan but reflected and shifted.
cot x 的周期为 π，在 sin x = 0 处有垂直渐近线，并在其间从 +∞ 递减到 –∞。其图像类似正切，但经过反射和平移。
6. Transformations of Trigonometric Graphs | 三角函数图像的变换
General sine or cosine function: f(x) = A sin(B(x – C)) + D. Here A is amplitude, period = 2π/|B| D is vertical shift. These parameters allow modelling of countless oscillatory phenomena.
一般正弦或余弦函数：f(x) = A sin(B(x – C)) + D。其中 A 为振幅，周期 = 2π/|B|，C 为水平（相位）平移，D 为垂直平移。这些参数使得无数振荡现象得以建模。
Vertical scaling by A stretches the graph vertically; if A > 1, peaks become taller; if 0 < A < 1, the wave is compressed. Negative A reflects across the x-axis.
垂直缩放 A 使图像纵向拉伸；若 A > 1，波峰变高；若 0 < A < 1，波形被压缩。负的 A 则产生关于 x 轴的反射。
B affects horizontal compression: larger |B| gives more cycles in a fixed interval. Phase shift C moves the curve left or right. D moves the midline up or down, changing the equilibrium position.
B 影响水平压缩：|B| 越大，固定区间内的周期数越多。相位移 C 使曲线左右移动。D 则将中线上下移动，改变平衡位置。
For tangent, f(x) = A tan(B(x – C)) + D has period = π/|B|, asymptotic lines shift accordingly, and vertical stretch changes steepness.
对于正切，f(x) = A tan(B(x – C)) + D 的周期为 π/|B|，渐近线相应移动，纵向拉伸则改变陡峭程度。
7. Period and Amplitude | 周期与振幅
nd cos, the fundamental period is 2π; for tan, it is π. After transformation, period = 2π/|B| for sine and cosine, π/|B| for tangent.
周期是使得对所有 x 都有 f(x + P) = f(x) 的最小正区间。对 sin 和 cos，基本周期为 2π；对 tan，为 π。变换后，sin 和 cos 的周期 = 2π/|B|，tan 的周期 = π/|B|。
Amplitude is half the difference between maximum and minimum values, defined only for bounded functions like sin and cos. It equals |A| in the general form. Tangent has no amplitude because its range is unbounded.
振幅是最大值与最小值之差的一半，只对 sin 和 cos 等有界函数定义。在一般形式中等于 |A|。正切函数因其值域无界而没有振幅。
Understanding these properties helps in sketching graphs quickly by marking key points: maxima, minima, zeroes, and midline intercepts.
理解这些性质有助于通过标记关键点（极大值、极小值、零点以及中线交点）快速绘制草图。
8. Symmetry and Even/Odd Properties | 对称性与奇偶性
Trigonometric functions display useful symmetries. Cosine is an even function: cos(–x) = cos x, symmetric about the y-axis. Sine and tangent are odd: sin(–x) = –sin x, tan(–x) = –tan x, symmetric about the origin.
三角函数呈现出有用的对称性。余弦是偶函数：cos(–x) = cos x，图像关于 y 轴对称。正弦和正切是奇函数：sin(–x) = –sin x，tan(–x) = –tan x，图像关于原点对称。
ation. For reciprocal functions, sec is even (since sec x = 1/cos x), csc and cot are odd.
这些奇偶性质可简化方程和积分。对于倒数函数，sec 是偶函数（因为 sec x = 1/cos x），csc 和 cot 是奇函数。
Complementary and supplementary angle identities (e.g., sin(π – x) = sin x, cos(π – x) = –cos x) also reflect symmetry about certain axes or points.
补角和余角恒等式（如 sin(π – x) = sin x，cos(π – x) = –cos x）也反映了关于特定轴或点的对称性。
9. Fundamental Trigonometric Identities | 基本三角恒等式
1. Dividing by cos²θ yields 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.
恒等式是对所有允许值都成立的方程。毕达哥拉斯恒等式最为基本：sin²θ + cos²θ = 1。除以 cos²θ 得 1 + tan²θ = sec²θ；除以 sin²θ 得 1 + cot²θ = csc²θ。
Double-angle formulas: sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ. These are used extensively in calculus and equation solving.
二倍角公式：sin(2θ) = 2 sin θ cos θ, cos(2θ) = cos²θ – sin²θ = 2 cos²θ – 1 = 1 – 2 sin²θ。这些在微积分和方程求解中广泛使用。
Compound angle formulas (addition/subtraction) and product-to-sum formulas also appear. In IB, the focus is on applying these to simplify expressions and prove relationships.
和差角公式以及积化和差公式也会出现。在IB中，重点在于应用这些公式化简表达式和证明关系。
Memorising and being able to derive these relationships is crucial for exam flexibility.
记忆并能够推导这些关系对于考试中的灵活应对至关重要。
10. Solving Trigonometric Equations | 解三角方程
circle and periodicity. The general solution for sin x = a is x = arcsin(a) + 2nπ or x = π – arcsin(a) + 2nπ, n ∈ ℤ.
求解诸如 sin x = k、cos x = k 或 tan x = k 的方程，通常在一个给定区间内，依赖于单位圆和周期性的知识。sin x = a 的通解为 x = arcsin(a) + 2nπ 或 x = π – arcsin(a) + 2nπ，n ∈ ℤ。
of the inverse trigonometric functions (arcsin, arccos, arctan) gives the principal value.
对于 cos x = a：x = ± arccos(a) + 2nπ。对于 tan x = a：x = arctan(a) + nπ。使用反三角函数（arcsin、arccos、arctan）给出主值。
Equations often need simplification using identities to a basic form. For example, 2 sin²x – quadratic in cos x. Factoring and checking for extraneous solutions are important skills.
方程常常需要用恒等式化简成基本形式。例如，2 sin²x – cos x = 1 可化为关于 cos x 的二次方程。因式分解并检查增根是重要技能。
Always verify solutions within the specified interval and consider the periodic nature to include all possible answers.
务必在指定区间内验证解，并考虑周期性以包含所有可能答案。
11. Modelling with Trigonometric Functions | 三角函数建模
rature variations, sound waves, and seasonal patterns. A typical model is f(t) = A sin(B(t – ves amplitude, D gives the average value, period = 2π/B corresponds to the cycle duration.
三角函数模型描述周期性现象：潮汐、温度变化、声波和季节模式。典型模型为 f(t) = A sin(B(t – C)) + D，其中 t 为时间。A 给出振幅，D 给出平均值，周期 = 2π/B 对应循环时长。
By interpreting real data, students identify maximum/minimum points, period, and phase shift. They then construct the function and use it to predict future values or times when a certain level is reached.
通过解读实际数据，学生识别最大/最小值点、周期和相位移。然后构建函数，并用它预测未来值或达到特定水平的时刻。
This application links graphical transformations to practical contexts, reinforcing the importance of parameter meaning.
这一应用将图像变换与实际情境联系起来，强化了参数含义的重要性。
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