📚 Chapter 7 Review: Trigonometry | 第7章复习：三角学
In this chapter review, we consolidate the core trigonometric concepts tested in the Edexcel A-Level Mathematics course. From radian measure and key identities to solving equations and modelling periodic phenomena, a systematic grasp of these topics is essential for success. Each section below pairs the key idea with a worked-style explanation, first in English then in Chinese, so you can self-assess your understanding.
本章复习将系统梳理Edexcel A-Level数学课程中三角函数的核心考点。从弧度制与基本恒等式，到解三角方程和周期现象建模，扎实掌握这些知识是通过考试的关键。下面每一小节均以英文要点搭配中文解析的形式呈现，方便你自我检测理解程度。
1. Radian Measure | 弧度制
n, 1 radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. The conversion factor is 180° = π rad, so to convert degrees to radians multiply by π/180, and to convert radians to degrees multiply by 180/π. For A-Level problems, you must set your calculator to radian mode whenever the question uses π or when dealing with any calculus involving trigonometric functions.
弧度制是在进行微分、积分或使用弧长、扇形面积公式时表达角度的自然方式。1弧度定义为半径长度的圆弧所对的圆心角。换算关系为180° = π rad，因此度数转弧度乘π/180，弧度转度数乘180/π。在A-Level解题中，只要题目涉及π或任何与三角函数有关的微积分计算，都必须将计算器设置为弧度模式。
- π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°
- 180° = π, 360° = 2π
- Make sure your calculator shows R or RAD.
2. Arc Length and Sector Area | 弧长与扇形面积
= (1/2)r²θ. For a segment, subtract the triangular area from the sector area: A_segment = (1/2)r²(θ − sin θ). Many exam questions combine these formulas with triangle geometry or ask you to find perimeter and shaded areas.
对于半径为r的圆，弧长s和扇形面积A与以弧度计量的圆心角θ成正比。公式为s = rθ，A = (1/2)r²θ。对于弓形，从扇形面积中减去三角形面积：A_弓形 = (1/2)r²(θ − sin θ)。许多考题会将上述公式与三角形几何结合起来，要求计算周长或阴影部分面积。
s = rθ, Asector = ½ r²θ
Always check whether θ is in radians before applying these formulas; if the angle is given in degrees, convert it first.
3. Basic Trigonometric Identities | 基本三角恒等式
csc²θ. These are essential when simplifying expressions, proving identities, and solving equations where you need to rewrite everything in terms of sine and cosine.
两个基本恒等式为 sin²θ + cos²θ = 1 以及 tan θ = sin θ / cos θ。由此可推导出其他形式，如1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。在化简表达式、证明恒等式以及需要将所有项化为正弦和余弦进行求解的方程中，这些恒等式不可或缺。
sin²θ + cos²θ ≡ 1, tan θ ≡ sin θ / cos θ
When proving identities, start from the more complicated side and use the identities to reduce it to the simpler side.
4. Trigonometric Graphs | 三角函数图像
l as transformations such as y = a sin(bx + c) + d. For sine and cosine, the period is 2π/|b|, amplitude is |a|, and phase shift is −c/b. Tangent has period π/|b|. Labelling the axes with key points – intercepts, maxima, minima, and asymptotes – is regularly demanded in exam curve-sketching questions.
你必须能够画出y = sin x、y = cos x和y = tan x的图像，以及类似y = a sin(bx + c) + d的变换图像。对于正弦和余弦，周期为2π/|b|，振幅为|a|，相位平移为−c/b。正切函数的周期为π/|b|。在图像题中，需要标明关键点——截距、最值点和渐近线。
- y = sin x: max 1, min −1, period 2π
- y = cos x: max 1, min −1, period 2π, shifted left by π/2 relative to sine
- y = tan x: asymptotes at x = π/2 + nπ
5. Solving Trigonometric Equations | 解三角方程
cle or the symmetry properties of the graph to find all solutions in the given interval. Always factorise where possible – for example, 2 sin x cos x − sin x = 0 becomes sin x (2 cos x − 1) = 0 – ble expression unless you consider the zero case. For quadratic in form, use substitution.
解形如sin θ = 0.5（0 ≤ θ < 2π）的方程时，首先用计算器反函数求出主值，然后借助单位圆或图像的对称性找出区间内所有解。务必在可能因式分解时进行分解——例如2 sin x cos x − sin x = 0化为sin x (2 cos x − 1) = 0——切忌随意约去含未知量的公因式，必须考虑其可能为零的情况。对于二次型方程，可采用换元法。
sin θ = k → θ = sin⁻¹k, π − sin⁻¹k (and add 2πn)
6. Compound Angle Formulae | 和角公式
t know how to apply them. sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; and tan(A ± B) = (tan A ± tan B) / (1 ∓ m technique is to use these formulas to find exact values, e.g., sin 75° = sin(45° + 30°).
和角公式可以将sin(A ± B)、cos(A ± B)和tan(A ± B)展开。这些公式已列在A-Level公式手册中，但你必须能够熟练运用。sin(A ± B) = sin A cos B ± cos A sin B；cos(A ± B) = cos A cos B ∓ sin A sin B；tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)。一个非常常见的考题是利用这些公式求精确值，例如sin 75° = sin(45° + 30°)。
sin(A + B) = sin A cos B + cos A sin B
7. Double Angle Formulae | 二倍角公式
²θ − 1 = 1 − 2 sin²θ. The form you choose usually depends on the other terms in the equation. For sin 2θ, the formula is sin 2θ = 2 sin θ cos θ, and for tan 2θ it' are heavily used in integration, derivation of further identities, and solving equations.
二倍角公式是与A = B对应的和角公式特例。你需要记住cos 2θ的三种形式：cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。选择哪种形式往往取决于方程中的其他项。sin 2θ的公式为sin 2θ = 2 sin θ cos θ，tan 2θ的公式为tan 2θ = 2 tan θ / (1 − tan²θ)。二倍角公式在积分、推导其他恒等式以及解方程中被大量使用。
cos 2θ ≡ 1 − 2 sin²θ (useful when integrating sin²θ)
8. R-Formulas (Harmonic Form) | R公式（调和形式）
ns of the form a sin θ + b cos θ can be written as R sin(θ + α) or R cos(θ − α), where R = √a² + b² n form). This technique is key for finding maximum/minimum values and solving equations where sin θ and cos θ appear linearly. The exam often asks for the maximum value of the expression and the smallest positive θ for which it occurs.
形如a sin θ + b cos θ的表达式可改写为R sin(θ + α)或R cos(θ − α)，其中R = √a² + b²，tan α = b/a（取决于所选形式）。在求最大/最小值以及解同时含有sin θ和cos θ线性组合的方程时，这一方法尤为关键。考试常要求找出该表达式的最大值以及该值首次出现时的最小正角θ。
a sin θ + b cos θ ≡ R sin(θ + α), R = √a² + b², tan α = b/a
Always indicate the quadrant of α by considering the signs of a and b.
9. Modelling with Trigonometric Functions | 三角函数的建模应用
Many real-world scenarios – seasonal temperature change, tides, height of a Ferris wheel – period = 2π/B), C is horizontal shift, and D is the vertical shift (average value). You may be asked to interpret parameters, find times at which a certain height is reached, or solve inequalities using graphs and algebra.
许多实际情境——如季节性气温变化、潮汐、摩天轮高度——都可以用正弦或余弦曲线建模。典型的模型是H(t) = A sin(B(t − C)) + D 或 H(t) = A cos(B(t − C)) + D，其中A为振幅，B控制周期（周期 = 2π/B），C为水平平移量，D为垂直平移量（平均值）。考题可能要求解释参数、计算达到某一高度的时刻，或结合图像和代数方法解不等式。
- Period = 2π/B; frequency = B/(2π)
- Ensure time units match the angle unit (always radians in calculus-based modelling).
10. Small Angle Approximations | 小角近似
These approximations are derived from the Maclaurin series and are extremely useful for simplifying expressions in physics and engineering contexts within maths problems. Exam questions may ask you to prove the approximations using geometric arguments or to use them to evaluate limits.
当θ很小且以弧度为单位时，sin θ ≈ θ，cos θ ≈ 1 − θ²/2，tan θ ≈ θ。这些近似源自麦克劳林级数，在数学题中的物理和工程背景下用于化简表达式非常有效。考题可能要求利用几何论证来证明这些近似，或利用它们计算极限。
sin 0.1 ≈ 0.1, cos 0.1 ≈ 1 − (0.1)²/2 = 0.995
Remember that the error in sine and tangent approximations is of order θ³, while for cosine it is θ⁴.
11. Trigonometric Integration and Differentiation | 三角微积分
in x, d/dx (tan x) = sec² x. The integrals are: ∫ cos x dx = sin x + C, ∫ sin x dx = −cos x + C, ∫ sec² x dx = tan x + C. For integrating sin² x or cos² x, use the double-angle identities to rewrite them in terms of cos 2x. For products like sin³ x or sin x cos x, use substitution or identities.
你必须熟练地掌握标准三角函数的微分和积分。导数为：d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。积分公式为：∫ cos x dx = sin x + C，∫ sin x dx = −cos x + C，∫ sec² x dx = tan x + C。对于积分sin² x或cos² x，需要利用二倍角公式将其改写为含cos 2x的形式。对于sin³ x或sin x cos x等的乘积形式，可用换元法或恒等式处理。
∫ sin² x dx = ∫ (1 − cos 2x)/2 dx = x/2 − (sin 2x)/4 + C
12. Chapter Summary and Tips | 章节总结与备考技巧
s for calculus, double-check your calculator mode, and sketch a graph whenever you are stuck on an equation-solving problem. Practise proving identities daily, and time yourself on mixed exercises that combine sector area, trigonometric equations, and modelling. When revising, create a personal one-page formula summary focusing on the connections: how compound angles lead to double angles, how sin²θ + cos²θ = 1 underpins most proofs, and how R-formulas unify sine and cosine terms.
掌握Edexcel A-Level三角学不仅需要记忆公式，更要理解各部分之间的联系。凡是涉及微积分都必须采用弧度制，随时检查计算器模式，在解方程卡壳时尝试画图辅助。每天坚持练习证明恒等式，并在限时条件下完成涵盖扇形面积、三角方程和建模的综合性习题。复习时制作一页个性化的公式总结，聚焦其中的联系：和角公式如何推出二倍角公式，sin²θ + cos²θ = 1如何支撑大多数证明，以及R公式如何统一正弦与余弦项。
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