📚 Using Trigonometric Models | 使用三角函数模型
Trigonometric models, built from sine and cosine functions, are powerful tools for representing and predicting cyclic behaviour in the world around us. From the ebb and flow of ocean tides to the oscillation of a pendulum, these models capture periodic patterns with remarkable precision. In the IB Mathematics curriculum, mastering trigonometric models means understanding how to interpret parameters, fit a function to real data, and solve equations that reveal critical moments such as maximum height, equilibrium, or crossing a threshold.
三角函数模型基于正弦和余弦函数建立，是描述和预测周期性现象的有力工具。从海洋潮汐的涨落到钟摆的摆动，这些模型精准地捕捉了循环模式。在IB数学课程中，掌握三角函数模型意味着理解如何解释参数、将函数与实际数据拟合，并求解方程以揭示诸如最大高度、平衡位置或超过阈值等关键时刻。
1. Introduction to Trigonometric Models | 三角函数模型简介
A trigonometric model is simply a sine or cosine function whose amplitude, period, phase shift, and vertical shift have been adjusted to match a real-world pattern. Because these four features completely describe a sinusoidal wave, we can tailor them to represent nearly any phenomenon that repeats at regular intervals — temperature variations over a year, blood pressure cycles, or the voltage of alternating current.
三角函数模型本质上是一个经过振幅、周期、相位移动和垂直平移调整的正弦或余弦函数，使其贴合某种现实规律。由于这四个特征可完全刻画正弦波，我们便能将其调整来描述几乎任何有规律重复的现象——如年气温变化、血压波动或交流电电压。
2. The General Sine and Cosine Functions | 一般正弦和余弦函数
The foundation of every trigonometric model is the general form: y = A sin(B(x − C)) + D or y = A cos(B(x − C)) + D ycle begins at the midline and increases, sine is natural; if it starts at a maximum or minimum, cosine is usually more convenient.
每个三角函数模型的基础都是一般形式：y = A sin(B(x − C)) + D 或 y = A cos(B(x − C)) + D。其中A控制振幅，B决定周期，C表示水平位移（相位移动），D确定了垂直平移。在x = 0处选择正弦或余弦通常取决于实际情况：如果循环从中线开始并向上升，用正弦很自然；如果循环从最大值或最小值开始，余弦往往更方便。
3. Understanding Amplitude and Vertical Shift | 理解振幅与垂直平移
Amplitude |A| dicates a reflection across the midline if negative, but amplitude itself is always taken as an absolute value when describing the wave height.
振幅|A|衡量的是最大值与最小值之间距离的一半。垂直平移D则是最大值与最小值的平均值。对于在35和15之间波动的模型，振幅 = (35 − 15)/2 = 10，垂直平移 = (35 + 15)/2 = 25。A的符号若为负则表示沿中线镜像翻转，但在描述波高时振幅本身总取绝对值。
4. Period and Frequency: The Role of B | 周期与频率：参数B的作用
Parameter B compresses or stretches the graph horizontally. For functions in radians, the period is 2π/|B|. If the period is known — say 12 hours for a tidal cycle — we can solve |B| = 2π/12 = π/6. In degree mode the period becomes 360°/|B| reciprocal of the period. A smaller period means a higher frequency and a larger B value.
参数B在水平方向上压缩或拉伸图像。对于使用弧度的函数，周期为 2π/|B|。若已知周期——例如潮汐周期为12小时——我们可解出 |B| = 2π/12 = π/6。在角度制下周期为 360°/|B|。频率在物理情境中常用，是周期的倒数。周期越小意味着频率越高，B值也越大。
5. Phase Shift and Horizontal Translation | 相位移动与水平平移
Phase shift moves the wave left or right. In the form sin(B(x − C)), the shift is C units to the right; a plus sign inside the bracket, sin(B(x + C)), indicates a shift to the left. Phase shift is especially important when modelling events that do not start at the typical zero of sine or cosine. For example, if the highest temperature of a day occurs at 14:00 rather than at noon, a phase shift adjusts the peak position correctly.
相位移动将波形向左或向右平移。在形式为 sin(B(x − C)) 的表达式中，平移为向右移动 C 个单位；括号内为加号，sin(B(x + C))，则表示向左移动。当所建模的事件并非起始于正弦或余弦的典型零点时，相位移动尤其重要。例如，若一天的最高气温出现在14:00而非正午，则需要通过相位移动来正确调整峰值位置。
6. Modeling Real-World Periodic Phenomena | 建模现实世界的周期性现象
er t minutes can be modelled by h(t) = −25 cos((2π/30)t) + 28, assuming boarding at the lowest point at t = 0. A negative cosine places the rider at the minimum at t = 0 and then rising.
建立模型通常从识别数据的最大值、最小值、周期以及起点的变化情况开始。一座摩天轮半径为25米，登车平台距地面3米，旋转一圈耗时30分钟，那么t分钟后高度h(t)（米）可以用 h(t) = −25 cos((2π/30)t) + 28 来模型化，这里假设t = 0时在最低点登车。使用负余弦可让乘客在t = 0时处于最低点并随后上升。
7. Determining Model Parameters from Data | 从数据确定模型参数
Given a data table or a list of conditions, follow a systematic approach:
- D: average of max and min values.
- A: half the difference between max and min.
- B: compute 2π/period (radians) or 360°/period (degrees).
- C: use a known point to solve for the horizontal shift. Often substitute a coordinate where the function crosses the midline on an upward or downward trend, depending on whether sine or cosine is used.
在给出数据表格或条件列表时，可遵循一套系统步骤：
- D：最大值与最小值的平均值。
- A：最大值与最小值之差的一半。
- B：计算 2π/周期（弧度）或 360°/周期（角度）。
- C：利用一个已知点求解水平位移。通常代入函数以向上或向下趋势穿过中线的坐标，具体取决于使用的是正弦还是余弦。
8. Solving Equations and Interpreting Solutions | 解方程并解释解的意义
n a certain level is reached. Because trigonometric functions are periodic, we obtain a primary solution plus integer multiples of the period. Always consider the domain given in the problem and check that all solutions are physically meaningful. For instance, a temperature model might give two solutions for a warm-enough condition — one in the morning and one in the afternoon — both valid within a 24-hour interval.
模型建立后，我们常需要求解 h(t) = k 以确定何时达到某一水平。由于三角函数具有周期性，我们会得到主解再加上周期的整数倍。务必考虑题目中给出的定义域，并检查所有解在物理上是否合理。例如，温度模型可能针对“足够暖和”这一条件给出两个解——一个在上午，一个在下午——两者在24小时区间内都有效。
9. Advanced Example: Tidal Heights | 进阶案例：潮汐高度
ccurs at t = 2 (02:00). To find when the depth is exactly 10 m, we solve 5 cos(πt/6 − π/3) + 8 = 10, leading to cos(πt/6 − π/3) = 0.4. This generates multiple times each day, illustrating the value of the model for harbour operations.
某港口的深度可以用 d(t) = 5 cos(πt/6 − π/3) + 8 来模型化，其中d以米为单位，t为午夜后的小时数。振幅为5米，因此水深在3米至13米间变化。周期为 2π ÷ (π/6) = 12 小时。相位移动为 (π/3) ÷ (π/6) = 2 小时向右，即第一次高潮出现在t = 2（02:00）。若要找出深度恰为10米的时刻，解方程 5 cos(πt/6 − π/3) + 8 = 10，得到 cos(πt/6 − π/3) = 0.4。每天会产生多个解，展现了该模型对港口运作的价值。
10. Using Technology (GDC) for Regression | 使用技术（图形计算器）进行回归
IB exam problems often provide a table of data and ask for a sine regression using a graphical display calculator (GDC). After entering the data into lists, you can perform a SinReg to obtain parameters A, B, C, D directly. Always round to a suitable number of significant figures and be prepared to compare the regression model with a manually derived one. Remember to set the calculator to radian or degree mode according to the context, as a mismatch is a frequent source of error.
IB考题常常提供数据表格，要求利用图形计算器（GDC）进行正弦回归。将数据输入列表后，可执行正弦回归（SinReg）直接得到A, B, C, D参数。始终将值舍入到合适的有效数字位数，并准备好将回归模型与人工导出的模型进行比较。切记根据上下文将计算器设置为弧度或角度模式，模式不一致是常见错误根源。
11. Common Mistakes and How to Avoid Them | 常见错误及如何避免
also sometimes confuse the period and the coefficient B, writing period = 2πB instead of 2π/B. Finally, always check whether the question expects an exact form (in terms of π) or an approximate decimal.
一个典型错误是书写 B(x - C) 时遗漏括号，导致计算器将表达式理解为 B × (x - C) 而非 B 与整个(x - C)的乘积。另一个常见问题是使用反三角函数求解C时没有正确找到主值域。同学们有时还会混淆周期与系数B，将周期写作 2πB 而非 2π/B。最后，务必检查题目是要求精确形式（用π表示）还是近似小数。
12. Summary and Exam Tips | 总结与考试技巧
Trigonometric modelling ties together graphical understanding, algebraic manipulation, and real-life interpretation. When approaching a modelling question, first sketch a rough graph and label the midline, maximum, minimum, and key points. Then systematically extract A, D, B, C. If a regression is required, write down the model with rounded coefficients and use it for subsequent predictions. Always validate your model at a known point to catch arithmetic slips. With practice, the process becomes intuitive and offers a satisfying bridge between pure mathematics and the physical world.
三角函数建模将图像理解、代数运算和现实解读融为一体。处理建模问题时，先画一张粗略草图，标出中线、最大值、最小值和关键点。随后系统地提取A, D, B, C参数。若需回归，写出带舍入系数的模型并用其进行后续预测。始终在一个已知点上验证模型以发现算术错误。通过练习，这一过程会变得直观，在纯数学与物理世界之间架起一座令人满足的桥梁。
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图