📚 13E Sinusoidal Models | 13E 正弦模型
Sinusoidal models use sine or cosine functions to describe periodic phenomena such as tides, temperature fluctuations, sound waves, and ferris wheel motion. In IB Mathematics, sinusoidal modelling connects abstract trigonometric parameters to real-world behaviour, preparing students to interpret, construct, and refine mathematical models based on data.
正弦模型利用正弦或余弦函数来描述周期性现象，例如潮汐、温度变化、声波和摩天轮运动。在IB数学中，正弦建模将抽象的三角函数参数与现实世界的行为联系起来，帮助学生解读、构建并根据数据改进数学模型。
1. What Are Sinusoidal Models? | 什么是正弦模型？
A sinusoidal model is an equation of the form y = A sin(B(x – C)) + D or y = A cos(B(x – C)) + D that captures cyclic patterns. The term 'sinusoidal' wave-like shape that repeats at regular intervals, whether it is based on sine or cosine.
正弦模型是指形如 y = A sin(B(x – C)) + D 或 y = A cos(B(x – C)) + D 的方程，用于描述循环模式。“正弦型”一词指任何具有平滑波浪形状并以固定间隔重复出现的曲线，无论它基于正弦还是余弦。
These models arise naturally when a quantity oscillates around a central value. The key advantage is that once the four parameters A, B, C, D are determined, the entire wave is fully characterised, enabling prediction and analysis.
当某个量围绕中心值振荡时，这类模型自然会显现。关键优势在于，一旦确定了A、B、C、D四个参数，整个波形就完全明确了，从而能够进行预测和分析。
2. Quick Review of Sine and Cosine | 正弦与余弦函数回顾
The parent sine function y = sin(x) has a maximum value of 1, a minimum of – of 0. The cosine function is horizontally shifted relative to sine: cos(x) = sin(x + π/2).
基准正弦函数 y = sin(x) 的最大值为1，最小值为–1，每 2π 弧度（或 360°）重复一次，均值为0。余弦函数相对于正弦有水平位移：cos(x) = sin(x + π/2)。
Both functions have the same amplitude (1), the same period (2π), and are bounded between –1 and 1. When modelling real-world data, we scale and shift these base waves to match observed oscillations.
两个函数具有相同的振幅（1）、相同的周期（2π），并在–1到1之间变化。在对现实世界数据建模时，我们对这些基础波进行缩放和平移，以匹配观测到的振荡。
3. The General Sinusoidal Equation | 一般正弦方程
The most common form used in IB modelling tasks is y = A sin(B(x – en time) and y is the dependent variable. Cosine is used when data peaks align with x = 0.
IB建模任务中最常用的形式是 y = A sin(B(x – C)) + D，其中 x 是自变量（通常为时间），y 是因变量。当数据峰值出现在 x = 0 时，则常使用余弦形式。
This equation can also be written as y = A sin(Bx + φ) + D, where φ = – is the phase shift in a different format. Both equivalent representations appear in exams.
该方程也可以写成 y = A sin(Bx + φ) + D，其中 φ = –BC 是一种不同形式的相移。这两种等价表示在考试中都会出现。
4. Amplitude and Vertical Shift | 振幅与垂直位移
The amplitude |A| e midline. It is calculated as half the difference between the maximum and minimum values: |A| = (max – min)/2.
振幅 |A| 控制波形在中心线上下起伏的高度，计算公式为最大值与最小值差的一半：|A| = (max – min)/2。
inimum: D = (max + min)/2. In temperature models, D represents the mean daily temperature.
垂直位移 D 将中心线向上或向下移动。它是最大值与最小值的平均值：D = (max + min)/2。在温度模型中，D 代表日均温度。
5. Period and Frequency | 周期与频率
period is the horizontal length of one complete cycle. When x is in radians, period = 2π/|B|; when degrees are used, period = 360°/|B|. A small |B| stretches the wave, while a large |B| compresses it.
周期是一个完整循环的水平长度。当 x 以弧度表示时，周期 = 2π/|B|；若使用角度制，则周期 = 360°/|B|。较小的 |B| 会拉伸波形，较大的 |B| 则压缩波形。
Frequency (cycles per unit of x) is the reciprocal of the period: f = |B| B questions, students must extract B from a given period or time between successive peaks.
频率（每单位 x 的周期数）是周期的倒数：f = |B|/2π。在许多 IB 试题中，学生需要根据已知周期或相邻峰值之间的时间求出 B。
6. Phase Shift (Horizontal Translation) | 相移（水平平移）
The parameter C represents the horizontal shift of the wave. For y = sin(B(x – graph of the basic sine function is shifted right by C units if C > 0, and left if C < 0.
参数 C 表示波形的水平位移。对于 y = sin(B(x – C)) + D，当 C > 0 时，基础正弦函数的图像向右平移 C 个单位；当 C < 0 时，向左平移。
When modelling, C is chosen so that a key feature – such as a peak, trough, or rising midline crossing – aligns with the data. If using cosine, C will differ from the sine-based C for the same data.
建模时，选择 C 使得某个关键特征（如波峰、波谷或上升过中点）与数据对齐。如果使用余弦，相同数据的 C 值与正弦形式下的 C 会不同。
7. From Graph to Equation | 从图像到方程
ne if the curve passes through the midline upwards at x = C; otherwise adjust with cosine.
给定正弦图像，识别：(1) 中线 y = D, (2) 振幅 A, (3) 周期以求出 B, 以及 (4) 一个关键点以确定 C。若曲线在 x = C 处向上穿过中线，则使用正弦；否则可改用余弦进行调整。
e peaks 4 units apart and starts at a minimum at x = 0. D = (8+2)/2 = 5, A = 3, period = 4 ⇒ B = 2π/4 = π/2. Since it starts at a minimum, a reflected cosine is ideal: y = –3 cos(π/2 x) + 5.
例题：某图像在 8 和 2 之间振荡，相邻波峰相隔 4 个单位，且从 x = 0 处的最低点开始。D = (8+2)/2 = 5, A = 3, 周期 = 4 ⇒ B = 2π/4 = π/2。由于从最低点开始，适合使用反射的余弦：y = –3 cos(π/2 x) + 5。
8. Building a Model from Data: Manual Fitting | 从数据手动拟合模型
With a small table of values, estimate D and A from the average and spread. Determine the period by measuring the difference in x between two consecutive maxima or minima. Then select C so that the wave matches a reference point, e.g. the first maximum.
有了小规模数据表，可以从平均值和范围估算出 D 和 A。通过测量相邻两个最大值或最小值的 x 之差来确定周期。然后选择 C，使波形与一个参考点（如第一个最大值）匹配。
Manual fitting produces an approximate model. You should then calculate residuals (actual – predicted) at data points to evaluate how well the model fits. Small residuals indicate a useful model, while systematic patterns suggest the need for phase adjustment or a different trigonometric function.
手动拟合给出一个近似模型。然后应在各数据点计算残差（实际值 – 预测值），以评估模型的拟合程度。残差较小表明模型可用，而系统性的模式则暗示需要调整相移或选用不同的三角函数。
9. Sinusoidal Regression Using Technology | 使用技术的正弦回归
Most IB examinations allow – and expect – . Always check that your calculator is in radian mode unless the context specifies degrees.
大多数IB考试允许并要求使用图形计算器（GDC）。正弦回归（SinReg）将 y = A sin(Bx + C) + D 拟合到数据。除非题目明确指定角度制，否则务必确保计算器处于弧度模式。
obtaining regression parameters, convert them to the required form, such as y = A sin(B(x – n requested and always plot the model against the data to visually verify goodness of fit.
得出回归参数后，通过因式分解将其转化为所需形式，如 y = A sin(B(x – C)) + D。按照要求的精确度报告参数，并始终绘制模型与数据的对比图以直观检验拟合优度。
10. Application: Modelling Daily Temperature | 应用：日气温建模
Suppose the temperature in a city can be modelled by T(t) = 15 sin(π/12 (t – s 2π/(π/12) = 24 hours, and the peak occurs at t = 14 (2 PM) as sin reaches 1 when π/12 (t – 8) = π/2.
假设某城市的气温可用 T(t) = 15 sin(π/12 (t – 8)) + 25 建模，T 以°C 为单位，t 是午夜后的时数。振幅 15 °C 表明昼夜温差大；D = 25 °C 是平均气温。周期为 2π/(π/12) = 24 小时，峰值出现在 t = 14（下午2点），因为此时 sin 达到 1，即 π/12 (t – 8) = π/2。
If the minimum temperature occurs at 4 AM, the model yields T(4) = 15 sin(π/12 (–4)) + 25 = 15 sin(–π/3) + 25 ≈ 15(– ich may match winter data. Adjusting C can realign the trough without changing the period.
若最低气温出现在凌晨 4 点，该模型得出 T(4) = 15 sin(π/12 (–4)) + 25 = 15 sin(–π/3) + 25 ≈ 15(–0.866) + 25 = 12.0 °C，这可能与冬季数据相符。调整 C 可以重新对齐波谷而不改变周期。
11. Application: Modelling Tidal Height | 应用：潮汐高度建模
Tides can be modelled by h(t) = A cos(B(t – is 12.42 hours, so B = 2π/12.42 ≈ 0.506 rad/h. Using cosine with peak at t=0 gives C = 0.
潮汐可以用 h(t) = A cos(B(t – C)) + D 建模。对于某港口，高潮 4.2 m 出现在 00:00 和 12:25，低潮 0.8 m 出现在 06:15，则 D = (4.2 + 0.8)/2 = 2.5，A = 1.7。两次高潮之间的周期为 12.42 小时，所以 B = 2π/12.42 ≈ 0.506 弧度/时。使用余弦并在 t=0 达到峰值，则 C = 0。
1.7(0.050) + 2.5 = 2.59 m. This value can be cross-checked with tide tables for accuracy.
模型 h(t) = 1.7 cos(0.506t) + 2.5 可以预测任何时刻的水位。例如，在凌晨 3 点，t=3, h(3) = 1.7 cos(1.518) + 2.5 ≈ 1.7(0.050) + 2.5 = 2.59 m。该数值可与潮汐表进行对比以检验准确性。
12. Common Pitfalls and Tips | 常见误区与提示
Pitfall 1: Using degrees instead of radians when the model involves π. If the period is 2π/B, B is in radians per unit x. Keep the calculator consistent.
误区一：在模型包含 π 时使用角度制而非弧度制。若周期为 2π/B，B 的单位为弧度每单位 x。应保持计算器设置一致。
Pitfall 2: Confusing the phase shift form. The expression B(x – C) means the shift is C, not –C. Double-check by substituting the intended peak x-value.
误区二：混淆相移形式。表达式 B(x – C) 意味着平移量为 C，而不是 –C。可代入预期的峰值 x 值进行验证。
Pitfall 3: Ignoring the vertical shift when reading amplitude directly from a graph that does not oscillate symmetrically about the x-axis. Always find the midline first.
误区三：当图像不以 x 轴为对称振荡时，直接从图中读取振幅而忽略垂直位移。务必先确定中线。
Tips: Draw a sketch, label key points, and verify with one or two data points. When using regression, choose an initial guess for the period to help the algorithm converge.
提示：绘制草图，标注关键点，并用一到两个数据点进行验证。使用回归时，为周期提供一个初始猜测值，以帮助算法收敛。
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