📚 Trigonometry and Circular Functions | 三角学与圆函数
Trigonometry extends far beyond right-angled triangles; it forms the backbone of periodic phenomena, wave motion and advanced geometry. In IB Mathematics, the study of circular functions unifies angle measure, coordinate geometry and algebraic modelling. This article covers radian measure, the unit circle, trigonometric graphs, identities, equations and practical applications, equipping you with a complete revision toolkit.
三角学远不止直角三角形，它是周期现象、波动和高等几何的支柱。在 IB 数学中，圆函数的学习将角度度量、坐标几何与代数建模融为一体。本文涵盖弧度制、单位圆、三角函数图像、恒等式、方程以及实际应用，为你提供完整的复习工具包。
1. Radian Measure and the Unit Circle | 弧度制与单位圆
elationship simplifies calculus and modelling, because derivative formulas for trigonometric functions hold only when the argument is in radians.
弧度制基于弧长提供了一种自然的角度度量方式。当弧长等于半径时，所对的圆心角为 1 弧度。圆周长为 2πr，因此一整转为 2π 弧度，从而得出 360° = 2π rad。这种关系简化了微积分和建模，因为三角函数的导数公式仅当变量以弧度表示时才成立。
The unit circle is a circle of radius 1 centred at the origin. Any point on its circumference can be written as (cos θ, sin θ), where θ is the angle measured anticlockwise from the positive x-axis. This definition generalises sine and cosine to all real numbers, not just acute angles, and reveals their periodic nature directly.
单位圆是圆心在原点、半径为 1 的圆。圆周上任一点可表示为 (cos θ, sin θ)，其中 θ 是从正 x 轴逆时针测量的角度。这一定义将正弦和余弦推广到所有实数，而不仅仅是锐角，并直接揭示了它们的周期性本质。
2. Defining Sine, Cosine and Tangent from the Unit Circle | 从单位圆定义正弦、余弦和正切
For any real angle θ, sine is the y‑ point where the terminal side of the angle intersects the unit circle, and cosine is the x‑ eometric interpretation makes the signs and symmetries of the functions immediately visible.
对于任意实数角 θ，正弦是该角终边与单位圆交点的 y 坐标，余弦是 x 坐标，即 (x, y) = (cos θ, sin θ)。然后正切定义为 tan θ = sin θ / cos θ，前提是 cos θ ≠ 0。这种几何解释使函数的符号和对称性一目了然。
The exact values of sine, cosine and tangent for standard angles such as 0, π/6, π/4, π/3 and π/2 can be read directly from the unit circle or from special right‑angled triangles. These values are essential for solving equations without a calculator.
标准角（如 0、π/6、π/4、π/3 和 π/2）的正弦、余弦和正切的精确值可直接从单位圆或特殊直角三角形中得出。这些值对于在没有计算器的情况下解方程至关重要。
| θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
3. The CAST Diagram and Signs of Trigonometric Functions | CAST 图与三角函数的符号
The signs of sine, cosine and tangent in the four quadrants can be remembered using the CAST diagram: starting from the fourth quadrant and moving anticlockwise, Cosine is positive (4th), All are positive (1st), Sine is positive (2nd), Tangent is positive (3rd). This mnemonic helps determine the sign of any trigonometric ratio quickly and is invaluable when solving equations.
四个象限中正弦、余弦和正切的符号可以用 CAST 图记忆：从第四象限逆时针方向依次为：余弦正 (C)、全正 (A)、正弦正 (S)、正切正 (T)。这个助记符号有助于快速确定任何三角比的符号，在解方程时非常宝贵。
For an angle θ measured from the positive x-axis, a related acute angle α can be found such that the magnitude of the function equals that of the corresponding acute angle. The sign is then decided by the quadrant in which θ lies.
对于从正 x 轴测量的角 θ，可以找到一个相关锐角 α，使得函数的大小等于该锐角的对应值，然后符号由 θ 所在象限决定。
4. Fundamental Trigonometric Identities | 基本三角恒等式
through by cos²θ yields 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ.
从单位圆推导出的两个最重要的恒等式是毕达哥拉斯恒等式和商恒等式。毕达哥拉斯恒等式为：对于任意角 θ，sin²θ + cos²θ = 1。这直接来自单位圆的方程 x² + y² = 1。两边同除以 cos²θ 可得 1 + tan²θ = sec²θ；同除以 sin²θ 可得 1 + cot²θ = csc²θ。
and cos(π/2 − θ) = sin θ are also fundamental, reflecting the symmetry of the unit circle.
商恒等式 tan θ = sin θ / cos θ 用正弦和余弦定义了正切。余角恒等式如 sin(π/2 − θ) = cos θ 和 cos(π/2 − θ) = sin θ 也是基本的，反映了单位圆的对称性。
5. Graphs of Sine, Cosine and Tangent | 正弦、余弦和正切的图像
nuous wave that oscillates between −1 and 1, with period 2π. It starts at the origin, rises to a maximum at π/2, returns to zero at π, reaches a minimum at 3π/2 and completes a full cycle at 2π. Its intercepts occur at integer multiples of π.
y = sin θ 的图像是一条光滑、连续的波形，在 −1 到 1 之间振荡，周期为 2π。它从原点出发，在 π/2 处达到最大值，在 π 处回到零，在 3π/2 处达到最小值，在 2π 处完成一个完整周期。其零点出现在 π 的整数倍处。
graph consists of repeating, increasing branches that approach positive infinity from the left of an asymptote and negative infinity from the right.
y = cos θ 的图像是相似的波形，但向左平移了 π/2；它从最大值 (0,1) 开始，在 2π 处完成一个周期。正切函数 y = tan θ 在 θ = π/2 + nπ 处有垂直渐近线，周期为 π。其图像由重复的、递增的分支组成，从渐近线左侧趋向正无穷，右侧趋向负无穷。
6. Transformations of Trigonometric Graphs | 三角图像的变换
translated using parameters in the general form y = a sin(b(θ − c)) + d. The amplitude is |a|, which scales the wave vertically. The period becomes 2π/|b|, compressing the graph horizontally if |b| > 1 and stretching it if 0 < |b| raph horizontally, while d shifts the entire wave vertically, altering the principal axis.
三角函数图像可以通过一般形式 y = a sin(b(θ − c)) + d 中的参数进行拉伸、压缩和平移。振幅为 |a|，在垂直方向上对波形进行缩放。周期变为 2π/|b|；当 |b| > 1 时图像水平压缩，当 0 < |b| < 1 时图像水平拉伸。相移 c 使图像水平移动，而 d 将整个波形垂直移动，改变主轴的位置。
Similar transformations apply to cosine and tangent, though the tangent period is π/|b|. When sketching transformed graphs, always identify amplitude, period, phase shift and vertical shift step by step. Labelling key points such as maxima, minima and intercepts ensures accuracy.
类似的变换也适用于余弦和正切，但正切的周期是 π/|b|。在绘制变换后的图像时，要逐步确定振幅、周期、相移和垂直平移。标注最大值、最小值和截距等关键点可确保准确性。
7. Inverse Trigonometric Functions | 反三角函数
Since sine, cosine and tangent are not one‑to‑one over their entire domains, their inverses are defined by restricting the domain. The function arcsin x (or sin⁻¹x) returns an angle in [−π/2, π/2] whose sine is x. Similarly, arccos x returns an angle in [0, π], and arctan x returns an angle in (−π/2, π/2). These principal value ranges are essential when solving equations or evaluating expressions.
由于正弦、余弦和正切在其整个定义域上不是一一对应的，因此通过限制定义域来定义它们的反函数。函数 arcsin x（或 sin⁻¹x）返回 [−π/2, π/2] 内正弦值为 x 的角。类似地，arccos x 返回 [0, π] 内的角，arctan x 返回 (−π/2, π/2) 内的角。在解方程或求值时，这些主值区间至关重要。
arly useful for finding general solutions to trigonometric equations, where one principal solution is found first and then combined with the periodicity of the original function.
反三角函数的图像是原函数受限部分关于直线 y = x 的反射。它们在寻找三角方程的通解时特别有用：首先求出一个主解，然后结合原函数的周期性得到全部解。
8. Solving Trigonometric Equations | 解三角方程
= π/6, and the second solution in [0, 2π] is θ = 5π/6, because sine is positive in the first and second quadrants.
解三角方程时，首先分离出三角函数，然后在指定区间（通常是 0 ≤ θ ≤ 2π）内确定满足方程的角度。利用单位圆或 CAST 图找出所有解。例如，若 sin θ = 1/2，主解为 θ = π/6，而在 [0, 2π] 内的第二个解为 θ = 5π/6，因为正弦在第一和第二象限均为正。
ite the range accordingly and solve for 2θ before halving the results. Always check that final solutions lie within the given interval. Quadratic equations in sin θ or cos θ can be solved by substitution and factoring.
当自变量是变量的倍数时（例如 sin(2θ) = 0.5），应相应扩大范围并先求解 2θ，然后将结果除以 2。务必检查最终解是否落在给定区间内。关于 sin θ 或 cos θ 的二次方程可通过换元和因式分解求解。
9. The Sine Rule and the Cosine Rule | 正弦定理与余弦定理
n C, where a, b, c are side lengths and A, B, C are the opposite angles. It is used when two angles and one side (AAS or ASA) are known, or when two sides and a non‑ncluded angle (SSA) are given, though the ambiguous case may yield two possible triangles.
正弦定理适用于任意三角形，表述为 a / sin A = b / sin B = c / sin C，其中 a、b、c 为边长，A、B、C 为对角。当已知两角一边 (AAS 或 ASA)，或已知两边及一个非夹角 (SSA) 时使用，但后者可能存在不定情况，可能产生两个三角形。
nown. Rearranging gives cos A = (b² + c² − a²) / (2bc), which is handy for finding angles.
余弦定理将三边与一角联系起来：a² = b² + c² − 2bc cos A。当已知两边及其夹角 (SAS) 或三边 (SSS) 时使用。移项得 cos A = (b² + c² − a²) / (2bc)，便于求角。
10. Area of a Triangle and Related Applications | 三角形面积及相关应用
formula is derived from the standard base × height formula and is especially useful in non‑right‑ area of a sector is ½ r²θ and the area of a triangle is subtracted to find a segment area.
三角形面积可表示为 ½ ab sin C，其中 a 和 b 是两边，C 是夹角。该公式由标准底乘高公式推导而来，在非直角三角形中尤为有用。它也出现在涉及圆弓形的问题中：扇形面积为 ½ r²θ，减去三角形面积即得弓形面积。
These area formulas extend to modelling real‑world situations such as calculating the area of a land plot given certain measurements and angles, or finding the swept area of a rotating radar beam. The ability to choose the correct rule and apply it efficiently is a key skill in IB examinations.
这些面积公式可扩展到实际建模中，例如根据给定的测量值和角度计算地块面积，或计算旋转雷达波束的扫掠面积。正确选择规则并高效运用是 IB 考试中的关键技能。
11. Modelling with Trigonometric Functions | 三角函数建模
− C)) + D, where A is amplitude, B determines period, C is horizontal shift and D is the equilibrium value.
许多周期现象，如潮汐、声波、钟摆运动和日温度变化，都可以用正弦或余弦函数建模。典型模型的形式为 f(t) = A sin(B(t − C)) + D 或 f(t) = A cos(B(t − C)) + D，其中 A 为振幅，B 决定周期，C 为水平平移，D 为平衡值。
When constructing a model, extract the amplitude from the difference between maximum and minimum values, determine the period from the time between successive peaks, and find the phase shift by aligning the graph with a known point. Questions often ask for the value of the function at a specific time, or the time at which a certain value first occurs.
构建模型时，从最大值和最小值之间的差提取振幅，根据连续峰值之间的时间确定周期，并通过将图像与已知点对齐来确定相移。题目常要求计算特定时间的函数值，或某一特定值首次出现的时间。
12. Summary and Exam Tips | 总结与应试技巧
Mastering trigonometry and circular functions requires fluency with the unit circle, exact values, identities and graph transformations. Regular practice of solving equations, both with and without a calculator, develops the speed and accuracy needed for exams. Always check the domain of solutions and whether the answer is required in radians or degrees.
掌握三角学与圆函数需要熟悉单位圆、精确值、恒等式和图像变换。定期练习解方程（无论是否使用计算器）可以培养考试所需的速度和准确性。始终检查解的定义域以及答案是要求用弧度还是角度表示。
When using the sine rule, watch for the ambiguous case (SSA); draw a diagram to visualise possible triangles. In modelling questions, label axes carefully and identify all parameters before substituting values. Finally, memorise the principal value ranges of inverse trigonometric functions—these are a common source of error.
使用正弦定理时，注意不定情况 (SSA)；画图以便直观了解可能的三角形。建模题中，仔细标注坐标轴，并在代入数值之前确定所有参数。最后，记住反三角函数的主值范围——这是常见的错误来源。
- Always convert angles to radians for calculus and graphing unless specified otherwise.
- 除非另有说明，微积分和绘图时始终将角度转换为弧度。
- Use the Pythagorean identity to simplify expressions before solving.
- 解题前先使用毕达哥拉斯恒等式化简表达式。
- Sketch a quick unit circle or CAST diagram to visualise sign and symmetry.
- 快速画出单位圆或 CAST 图以直观了解符号和对称性。
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