Modelling with Trigonometric Functions | 三角函数建模

📚 Modelling with Trigonometric Functions | 三角函数建模

Trigonometric models are used across Edexcel A-Level Mathematics to represent real-world situations that repeat in a regular cycle. This topic asks you to interpret the parameters of sine and cosine functions, build equations from given data, and solve the resulting equations within a specified time interval.

在爱德思 A-Level 数学中，三角函数模型用于表示以规律周期重复的现实情境。本专题要求你解读正弦和余弦函数中的参数、根据给定数据建立方程，并在指定时间区间内求解所得方程。

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1. Why Use Trigonometric Models? | 为什么使用三角函数模型？

Many quantities in mechanics, biology and geography are periodic: a Ferris wheel height, tide depth, temperature over 24 hours, alternating current, or the motion of a spring. A sine or cosine curve is the natural starting point because it has a constant period, a maximum, a minimum, and a smooth repeating shape.

力学、生物和地理中的许多量都具有周期性：摩天轮高度、潮汐深度、24 小时内的温度、交流电或弹簧运动。正弦或余弦曲线是自然的起点，因为它具有恒定周期、最大值、最小值以及平滑重复的形状。

In exam questions, a context will usually give you the maximum and minimum values, the period, and sometimes a starting value. Your job is to convert those facts into a trigonometric equation of the form y = a sin(b(x − c)) + d or y = a cos(b(x − c)) + d.

在考试题中，背景通常会给出最大值和最小值、周期，有时还会给出起始值。你的任务是将这些条件转化为形如 y = a sin(b(x − c)) + d 或 y = a cos(b(x − c)) + d 的三角函数方程。

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2. The Standard Sinusoidal Form | 标准正弦型函数

The two standard forms you should be confident with are y = a sin(b(x − c)) + d and y = a cos(b(x − c)) + d. The variable x is often replaced by t when modelling time. The same ideas apply to both sine and cosine.

你应该熟练掌握的两种标准形式是 y = a sin(b(x − c)) + d 和 y = a cos(b(x − c)) + d。当用于时间建模时，变量 x 通常用 t 代替。相同的方法也适用于正弦和余弦。

y = a sin(b(x − c)) + d

Period = 2π / b (radians) 或 360° / b (degrees)

Here, a is the amplitude, b is the angular frequency, c is the horizontal shift, and d is the vertical shift. The midline of the graph is y = d.

这里，a 是振幅，b 是角频率，c 是水平位移，d 是垂直位移。图像的中线为 y = d。

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3. Meaning of the Parameters | 参数的含义

The amplitude a is half the distance between the maximum and minimum values. It is always taken as a positive value when writing the model in the forms above. For a graph with maximum M and minimum m, use:

振幅 a 是最大值与最小值之间距离的一半。在写成上述形式时，a 通常取正值。对于最大值为 M、最小值为 m 的图像，使用：

a = (M − m) / 2

d = (M + m) / 2

The vertical shift d moves the whole curve up or down and gives the average value or midline. The parameter b changes the horizontal scale: as b increases, the period decreases. The parameter c shifts the graph horizontally; in y = a sin(b(x − c)) + d, a positive c shifts the curve to the right.

垂直位移 d 将整条曲线向上或向下移动，并给出平均值或中线。参数 b 改变水平尺度：b 增大，周期减小。参数 c 使图像水平移动；在 y = a sin(b(x − c)) + d 中，c 为正时曲线向右平移。

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4. Finding the Period and Frequency | 求周期与频率

If a context gives a period T, find b using the formula b = 2π / T when angles are measured in radians, or b = 360° / T when using degrees. For example, a tide with a period of 12.4 hours has b = 2π / 12.4 ≈ 0.5067 radians per hour.

如果题目给出周期 T，当角度以弧度表示时，使用公式 b = 2π / T 求出 b；当使用度数时，使用 b = 360° / T。例如，周期为 12.4 小时的潮汐，其 b = 2π / 12.4 ≈ 0.5067 弧度/小时。

T = 2π / b, f = 1 / T = b / 2π

Frequency f is the number of complete cycles per unit of time. In A-Level modelling, you will usually need the period rather than frequency, but the relationship above is useful when a question gives cycles per hour or cycles per second.

频率 f 是单位时间内完整周期的次数。在 A-Level 建模中，通常需要周期而不是频率，但当题目给出每小时或每秒的周期数时，上述关系非常有用。

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5. Choosing Sine or Cosine | 选择正弦还是余弦

Choosing between sine and cosine is a matter of the starting point. At x = 0, a sine curve without a phase shift starts at the midline and is increasing; a cosine curve without a phase shift starts at its maximum.

选择正弦还是余弦取决于起点。在 x = 0 处，没有相位移的正弦曲线从中线开始并向上增加；没有相位移的余弦曲线从最大值开始。

If a Ferris wheel passenger starts at the highest point, a cosine model is often simplest. If the passenger starts at the centre level and moves upwards, a sine model is more natural. You may still use a phase shift c to make either function fit the same data.

如果摩天轮乘客从最高点出发，通常用余弦模型最简单。如果乘客从中层水平位置开始并向上移动，则用正弦模型更自然。你仍然可以通过相位移 c 使任一函数拟合相同的数据。

Starts at maximum: y = a cos(bx) + d

Starts at midline increasing: y = a sin(bx) + d

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6. Building a Model from Data | 从数据建立模型

Suppose a quantity has maximum 15, minimum 5, and period 8. Compute a = (15 − 5) / 2 = 5, d = (15 + 5) / 2 = 10, and b = 2π / 8 = π / 4. If the quantity starts at the midline increasing, the model is:

假设某个量的最大值为 15，最小值为 5，周期为 8。计算 a = (15 − 5) / 2 = 5，d = (15 + 5) / 2 = 10，b = 2π / 8 = π / 4。如果该量从中线开始并增加，则模型为：

y = 5 sin(πx / 4) + 10

If the same quantity starts at its maximum when x = 0, replace sine by cosine:

如果同一个量在 x = 0 时从其最大值开始，则将正弦替换为余弦：

y = 5 cos(πx / 4) + 10

Always check the model at x = 0, at one quarter of the period, and at one full period. For y = 5 cos(πx / 4) + 10, at x = 0 we get 15; at x = 2 we get 10; at x = 4 we get 5; and at x = 8 we return to 15.

务必在 x = 0、四分之一周期和一个完整周期处检验模型。对于 y = 5 cos(πx / 4) + 10，当 x = 0 时得到 15；x = 2 时得到 10；x = 4 时得到 5；x = 8 时又回到 15。

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