📚 Trigonometry and Modelling | 三角函数与建模
Trigonometry and modelling is a core topic in Edexcel A‑Level Mathematics that unites graph transformations, trigonometric identities and real‑world periodic behaviour. It requires you to interpret amplitude, period, phase shift and vertical translation in order to construct and solve equations based on data such as tidal heights, temperature variations or mechanical oscillations.
三角函数与建模是 Edexcel A‑Level 数学的核心主题，它将图像变换、三角恒等式与现实世界的周期性行为结合在一起。你需要理解振幅、周期、相位移动和垂直平移，以便根据潮汐高度、温度变化或机械振动等数据建立并求解方程。
1. The foundations of trigonometric graphs | 三角函数图像的基础
The sine curve y = sin x is periodic with period 2π and amplitude 1, crossing the x‑axis at multiples of π. Its maximum value is 1 and minimum is –1, giving a smooth wave shape.
正弦曲线 y = sin x 的周期为 2π，振幅为 1，并在 π 的整数倍处穿过 x 轴。最大值为 1，最小值为 –1，呈现平滑的波浪形状。
π/2). The tangent function y = tan x has period π and vertical asymptotes at x = π/2 + nπ.
余弦图像 y = cos x 是正弦图像向左平移 π/2 得到的，因此 cos x = sin(x + π/2)。正切函数 y = tan x 的周期为 π，并在 x = π/2 + nπ 处有垂直渐近线。
2. Transformations of trigonometric functions | 三角函数图像的变换
stretches are controlled by a coefficient a in y = a sin x, which changes the amplitude to |a| orizontal stretches come from the coefficient b in y = sin(bx), giving a new period of 2π/|b|.
垂直伸缩由 y = a sin x 中的系数 a 控制，振幅变为 |a|。水平伸缩则由 y = sin(bx) 中的系数 b 决定，新周期为 2π/|b|。
A horizontal translation is introduced by c in y = sin(x + c), shifting the graph by – e up or down. Combining all four parameters yields the general form y = a sin(bx + c) + d.
y = sin(x + c) 中的 c 引起水平平移，将图像移动 –c 个单位；而 d 则使整条曲线向上或向下垂直移动。将四个参数组合起来就得到一般形式 y = a sin(bx + c) + d。
3. Modelling with y = a sin(bx + c) + d | 用 y = a sin(bx + c) + d 进行建模
ater in a harbour might be modelled by h(t) = a sin(bt + c) + d, with t measured in hours.
许多周期性情形都可以用这种正弦形式建模，其中 x 通常被替换为时间 t。例如，港口的水位高度可以用 h(t) = a sin(bt + c) + d 进行建模，t 以小时为单位。
The parameter a represents the amplitude, b determines the period (2π/b), c controls the horizontal shift, and d is the vertical shift representing the mean value. Understanding each component is essential for fitting the model to given data.
参数 a 代表振幅，b 决定周期 (2π/b)，c 控制水平移动，d 是表示平均值的垂直位移。理解每个组成部分对于将模型与给定数据拟合至关重要。
4. Determining amplitude and vertical shift | 确定振幅和垂直位移
nd minimum values from a data set or graph, the amplitude is half the difference: a = (max – min)/2. The vertical shift is the midpoint: d = (max + min)/2.
当从数据集或图像给出最大值和最小值时，振幅为差值的一半：a = (max – min)/2。垂直位移是中点：d = (max + min)/2。
For example, if a temperature oscillates between 5°C and 25°C, then a = (25 – t as T = 10 sin(bt + c) + 15, with b and c to be found from the period and starting point.
例如，若温度在 5°C 和 25°C 之间振荡，则 a = (25 – 5)/2 = 10，d = (25 + 5)/2 = 15。模型的初始形式为 T = 10 sin(bt + c) + 15，其中的 b 和 c 需要通过周期和起点来确定。
5. Period, frequency and the horizontal stretch | 周期、频率和水平伸缩
The period of y = a sin(bx + c) + d is 2π/|b| eriod is the time taken for one complete cycle. Frequency is the reciprocal of the period.
当 x 以弧度为单位时，y = a sin(bx + c) + d 的周期为 2π/|b|。如果自变量是时间 t，则周期是完成一个完整循环所需的时间。频率是周期的倒数。
To find b, set 2π/b equal to the observed period. For a 12‑ If the period is given in degrees, remember to convert to radians or work consistently in the required unit.
要求出 b，令 2π/b 等于观测到的周期。对于 12 小时的潮汐循环，2π/b = 12，因此 b = π/6。若周期以度数给出，请记住转换为弧度或保持单位一致。
6. Phase shift and horizontal translation | 相移和水平平移
on y = a sin(bx + c) can be rewritten as y = a sin[b(x + c/b)], so the horizontal shift is –c/b. If c is positive, the graph shifts to the left; if negative, to the right.
表达式 y = a sin(bx + c) 可改写为 y = a sin[b(x + c/b)]，因此水平平移量为 –c/b。若 c 为正，图像向左平移；若为负，则向右平移。
ine function starting at the origin is appropriate, so c may be 0 or adjusted accordingly.
在建模中，相移通常由一个已知的初始条件确定。如果在 t = 0 时水位正处于平均值并开始上升，那么从原点开始的正弦函数是合适的，此时 c 可以为 0，或根据需要进行调整。
7. Radian measure and its importance | 弧度制及其重要性
Radian measure is fundamental in trigonometric modelling because the derivative of sin x is cos x only when x is in radians. A‑Level problems almost always expect radian knowledge for calculus and equation solving.
弧度制在三角建模中至关重要，因为只有当 x 以弧度为单位时 sin x 的导数才是 cos x。A‑Level 题目几乎都要求使用弧度知识进行微积分和解方程。
ch from degrees to radians, multiply by π/180; from radians to degrees, multiply by 180/π. When a real‑orld model involves time in hours, the argument inside sine is purely a number in radians.
换算关系是 π rad = 180°。从度转为弧度，乘以 π/180；从弧度转为度，乘以 180/π。当真实世界的模型涉及以小时为单位的时间时，正弦内的参数是一个纯弧度数值。
8. Trigonometric identities for simplification | 用于化简的三角恒等式
/ cosθ and the derived forms 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are also essential.
恒等式 sin²θ + cos²θ = 1 经常用于在正弦和余弦的平方之间进行转换。正切恒等式 tanθ = sinθ / cosθ 以及派生形式 1 + tan²θ = sec²θ、1 + cot²θ = cosec²θ 也是必不可少的。
These identities allow you to reduce equations to a single trigonometric function, which is particularly helpful when separating variables or solving for an unknown in a model.
这些恒等式让你能够将方程化简为单一三角函数，在分离变量或求解模型中的未知量时特别有用。
9. Addition and double-angle formulas | 加法公式与倍角公式
addition formulas are sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. For tangent, tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
加法公式为 sin(A ± B) = sin A cos B ± cos A sin B 以及 cos(A ± B) = cos A cos B ∓ sin A sin B。对于正切，tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。
Setting B = A yields the double-angle formulas: sin 2A = 2 sin A cos A, cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A, and tan 2A = 2 tan A / (1 – tan²A). These are vital when transforming expressions like a sin x + b cos x.
令 B = A 可得到倍角公式：sin 2A = 2 sin A cos A，cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A，以及 tan 2A = 2 tan A / (1 – tan²A)。这些公式在转化如 a sin x + b cos x 的表达式时至关重要。
10. The form R sin(x ± α) and R cos(x ± α) | R sin(x ± α) 和 R cos(x ± α) 的形式
sin θ cos α + R cos θ sin α, then equate coefficients to get R cos α = a and R sin α = b.
形如 a sin θ + b cos θ 的表达式可以合并成一个单一的正弦或余弦函数。对于 R sin(θ + α)，展开得到 R sin θ cos α + R cos θ sin α，然后比较系数得 R cos α = a，R sin α = b。
R is found by R = √a² + b² uadrant chosen to match the signs of a and b. This form makes it easy to identify the maximum value R, the minimum –R, and to solve equations.
R 由 R = √a² + b² 给出，α 满足 tan α = b/a，并根据 a 和 b 的符号选择象限。这种形式便于确定最大值 R、最小值 –R，以及解方程。
a sin θ + b cos θ = R sin(θ + α), where R = √a² + b² and tan α = b/a, quadrant‑checked.
a sin θ + b cos θ = R sin(θ + α)，其中 R = √a² + b²，tan α = b/a，并检验象限。
11. Solving trigonometric equations in modelling contexts | 在建模情境下解三角方程
l to the target value, then rearrange into a basic trig equation, such as sin(bt + c) = k.
在建模问题中，你经常需要求出何时出现某一高度、温度或深度。令模型等于目标值，然后重组为基本三角方程，例如 sin(bt + c) = k。
nal answers in the required units. Radians are almost always used unless otherwise specified.
求解时先求出主值，然后加上周期或利用对称性。记得检查所给的定义域，如 0 ≤ t ≤ 24，并以要求的单位给出最终答案。除非特别说明，几乎总是使用弧度。
12. Exam tips and common pitfalls | 考试技巧与常见陷阱
Always read the question carefully to identify whether the independent variable is in degrees or radians, and stick to that unit throughout. Sketching a rough graph can help you visualise the period, amplitude and phase shift before writing equations.
务必仔细审题，明确自变量是用度还是弧度，并全程保持一致。在列出方程之前，画一个简图有助于直观把握周期、振幅和相移。
d from max/min values. Another is misinterpreting the sign of the phase shift: y = sin(2x – π) gives a shift of +π/2, not –π. Always check with your calculator in radian mode.
一个常见错误是在根据最大值/最小值计算 a 或 d 时忘记使用垂直位移。另一个是对相移符号的误解：y = sin(2x – π) 的平移量为 +π/2，而不是 –π。务必确保计算器处于弧度模式。
If you use the R sin(θ + α) method, watch the quadrant of α carefully; many marks are lost by selecting the wrong sign for α. Finally, present all angles in the required range and to the specified accuracy, often 1 decimal place or 3 significant figures.
如果使用 R sin(θ + α) 方法，请仔细判断 α 所在的象限；很多失分都是因为选错了 α 的符号。最后，所有角度都要落在要求的范围内，并保留指定的精度，通常是 1 位小数或 3 位有效数字。
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