📚 Edexcel A-Level Maths: Trigonometric Ratios and Equations | Edexcel A-Level 数学：三角比与三角方程
Trigonometric ratios and equations are central to the Edexcel A-Level Mathematics specification. They appear in pure mathematics questions on triangles, graphs, identities and real-world modelling. A secure command of sine, cosine, tangent, exact values and equation-solving is essential for high marks.
三角比与三角方程是 Edexcel A-Level 数学大纲的核心内容。它们在纯数学的三角形、图像、恒等式和实际问题建模中反复出现。熟练掌握正弦、余弦、正切、特殊角精确值以及解三角方程是取得高分的关键。
1. Where Trigonometry Appears in the Edexcel Specification | 三角学在 Edexcel 大纲中的位置
Trigonometry is examined in both years of the Edexcel A-Level Mathematics course. In Year 1 (Pure Mathematics 1) you meet the sine rule, the cosine rule, the area formula ½ ab sin C, exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90°, the graphs of the three functions, the identity sin² θ + cos² θ ≡ 1, the quotient identity tan θ ≡ sin θ / cos θ, and the solution of trigonometric equations in both degrees and radians. Radian measure, arc length and sector area also belong to this first year.
三角学在 Edexcel A-Level 数学两年的课程中都会被考查。第一年（纯数学 1）会学习正弦定理、余弦定理、面积公式 ½ ab sin C、0°、30°、45°、60°、90° 的正弦余弦正切精确值、三个函数的图像、恒等式 sin² θ + cos² θ ≡ 1、商数关系 tan θ ≡ sin θ / cos θ，以及用角度和弧度解三角方程。弧度制、弧长和扇形面积也属于第一年的内容。
In Year 2 (Pure Mathematics 2) the topic deepens: double-angle formulae, the harmonic form R sin(θ + α) or R cos(θ − α), more demanding identity work, and modelling problems where a periodic quantity such as temperature, tide height or the height of a Ferris wheel seat must be described by a trigonometric function. Questions are rarely a single step. A typical six-mark question may require you to extract a triangle from a diagram, choose the correct rule, substitute accurately, solve a further equation and then interpret the answer in context.
第二年（纯数学 2）会进一步深入：二倍角公式、辅助角形式 R sin(θ + α) 或 R cos(θ − α)、更复杂的恒等式变形，以及用三角函数描述温度、潮汐高度、摩天轮座舱高度等周期量的建模问题。这类题目很少只有一步。一道 6 分的典型题可能要求你从图形中提取三角形、选择正确的定理、准确代入、再解一个方程，最后结合情境解释答案。
One practical point at the very start: check whether your calculator is set to degrees or radians. Marks are regularly lost not because the method is wrong but because the mode does not match the question. If a question uses π, work in radians; if it uses the degree symbol, work in degrees.
有一个非常实际的起点：先检查计算器处于角度模式还是弧度模式。丢分往往不是方法错误，而是模式与题目不匹配。题目中出现 π 就用弧度，出现 ° 就用角度。
2. The Sine Rule | 正弦定理
For any triangle ABC, with side a opposite angle A, side b opposite angle B and side c opposite angle C, the sine rule states:
a / sin A = b / sin B = c / sin C
三角比与三角方程是 Edexcel A-Level 数学大纲的核心内容。对于任意三角形 ABC，边 a 对角 A，边 b 对角 B，边 c 对角 C，正弦定理为上式。
When you are finding a side, use the form with the sides on top. When you are finding an angle, flip the whole rule over and use the form with the sines on top:
sin A / a = sin B / b = sin C / c
求边时用“边在上”的形式；求角时把整个公式倒过来，用“正弦在上”的形式：
iguous case: because sin B = sin(180° − B), two possible triangles may exist. Only accept the second angle if it leaves a positive third angle, that is, if A + B < 180°.
当已知两角一边（AAS 或 ASA）或两边及一个非夹角（SSA）时使用正弦定理。在 SSA 情形中必须检查“不确定情形”：因为 sin B = sin(180° − B)，可能存在两个符合条件的三角形。只有当第二个角使第三个角仍为正，即 A + B < 180° 时，才接受它。
that the side opposite the larger angle is longer, which is a useful check on any answer.
例题：在三角形 ABC 中，A = 30°，B = 80°，a = 10 cm，求 b。由 b = a sin B / sin A = 10 × sin 80° / sin 30° = 10 × 0.9848 / 0.5 = 19.7 cm（保留三位有效数字）。注意大角对大边，这是检验答案是否合理的有效方法。
h give A + B < 180°, so two triangles are possible and both must be reported for full marks.
不确定情形例题：a = 5 cm，b = 8 cm，A = 30°。则 sin B = b sin A / a = 8 × 0.5 / 5 = 0.8，所以 B = 53.13° 或 B = 126.87°。两者都满足 A + B < 180°，因此存在两个三角形，必须都写出才能拿满分。
3. The Cosine Rule | 余弦定理
For any triangle ABC the cosine rule states:
a² = b² + c² − 2bc cos A
对于任意三角形 ABC，余弦定理为上式。
Use this version when you know two sides and the included angle (SAS) and want the third side. Remember that the side on the left must be the one opposite the angle you use on the right. If the labels differ, relabel carefully before substituting.
当已知两边及其夹角（SAS）要求第三边时使用这个形式。记住等号左边的边必须与右边所用的角相对。如果字母标注不同，代入前要小心重新标注。
To find an angle, rearrange the rule:
cos A = (b² + c² − a²) / (2bc)
要求角时，把公式变形为上式。
Use the rearranged version when you know all three sides (SSS). A quick sign check is valuable: if b² + c² is greater than a², then cos A is positive and A is acute; if b² + c² is less than a², then cos A is negative and A is obtuse. This immediately tells you whether your calculator answer of around 60° should actually be around 120° when the question is set in degrees.
当已知三边（SSS）求角时使用变形后的形式。做一次符号检查很有价值：若 b² + c² 大于 a²，则 cos A 为正，A 为锐角；若 b² + c² 小于 a²，则 cos A 为负，A 为钝角。这能立刻告诉你，当题目用角度制时，计算器给的约 60° 是否应该变成约 120°。
8.42 cm (3 s.f.). Round only at the final stage; premature rounding is one of the most common sources of lost accuracy marks.
例题：三角形两边为 7 cm、9 cm，夹角 62°。则 a² = 7² + 9² − 2 × 7 × 9 × cos 62° = 49 + 81 − 126 × 0.4695 = 130 − 59.15 = 70.85，所以 a = 8.42 cm（三位有效数字）。只在最后一步取近似值；过早取近似是最常见的精度失分原因之一。
4. Area of a Triangle | 三角形面积公式
The area of any triangle is given by:
Area = ½ ab sin C
任意三角形的面积由上式给出。
Here C must be the angle between sides a and b. This formula is used far more often than students expect, especially in questions that combine area with the sine and cosine rules, or in questions where a shape is split into two triangles and the total area must be found by adding the parts.
这里 C 必须是边 a 与边 b 的夹角。这个公式的使用频率远高于学生的预期，尤其在把面积与正弦定理、余弦定理结合的题目中，或是在把一个图形拆成两个三角形、需要用各部分面积相加来求总面积的题目中。
If the angle is obtuse, sin C is still positive, so the formula works without modification. If you are given three sides only, first use the cosine rule to find one angle, then apply the area formula. If you are given two angles and one side, first find the third angle, then use the sine rule to obtain the second side you need.
若角为钝角，sin C 仍为正，公式无需修改即可使用。若只给三边，先用余弦定理求出一个角，再套用面积公式。若给两角一边，先求出第三个角，再用正弦定理求出所需的第二条边。
1.6 cm² (3 s.f.). Always carry the unit; area units are squared, and a missing cm² is a genuine error in an examination.
例题：三角形中 a = 6 cm，b = 11 cm，夹角 C = 41°。面积 = ½ × 6 × 11 × sin 41° = 33 × 0.6561 = 21.6 cm²（三位有效数字）。一定要带单位；面积单位是平方，考试中漏写 cm² 属于实质性错误。
5. Choosing the Right Rule | 如何选择正确的定理
Most examination errors in this topic come from selecting the wrong rule rather than from arithmetic. Train yourself to identify the pattern of given information before you touch the calculator.
本主题中大多数考试错误来自选错定理，而不是算错。要训练自己在动计算器之前先识别已知信息的模式。
| Given information | 已知条件 | Method | 方法 |
| Two angles and one side (AAS, ASA) | 两角一边 | Sine rule | 正弦定理 |
| Two sides and a non-included angle (SSA) | 两边及非夹角 | Sine rule + ambiguous case check | 正弦定理 + 不确定情形检查 |
| Two sides and the included angle (SAS) | 两边及夹角 | Cosine rule or area rule | 余弦定理或面积公式 |
| Three sides (SSS) | 三边 | Cosine rule for an angle | 用余弦定理求角 |
| Right-angled triangle | 直角三角形 | SOHCAHTOA and Pythagoras | 三角函数比与勾股定理 |
A useful habit is to sketch the triangle, mark the given values in the correct positions, and label the unknown with a question mark. This makes the correct rule almost self-evident and also helps you decide whether the unknown is a side or an angle.
一个好习惯是把三角形画出来，把已知量标在对应位置，把未知量标上问号。这样正确定理几乎一目了然，也能帮你判断未知量是边还是角。
6. Exact Values, the Unit Circle and the CAST Diagram | 特殊角精确值、单位圆与 CAST 图
You are expected to know the exact values of sine, cosine and tangent for 0°, 30°, 45°, 60° and 90° without a calculator. These come from the two standard right-angled triangles: the half-square with angles 45°, 45°, 90° and the half-equilateral triangle with angles 30°, 60°, 90°.
你必须在不使用计算器的情况下记住 0°、30°、45°、60°、90° 的正弦、余弦、正切精确值。它们来自两个标准直角三角形：角为 45°、45°、90° 的半正方形，以及角为 30°、60°、90° 的半等边三角形。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | ½ | √2 / 2 | √3 / 2 | 1 |
| cos θ | 1 | √3 / 2 | √2 / 2 | ½ | 0 |
| tan θ | 0 | 1 / √3 | 1 | √3 | undefined | 无定义 |
The unit circle explains the signs. In the first quadrant all three ratios are positive; in the second only sine is positive; in the third only tangent is positive; in the fourth only cosine is positive. The CAST diagram encodes exactly this sequence, reading anticlockwise starting from the fourth quadrant.
单位圆解释了符号规律。第一象限三个比值全为正；第二象限只有正弦为正；第三象限只有正切为正；第四象限只有余弦为正。CAST 图正是对这一顺序的编码，从第四象限开始逆时针读取。
cos θ, tan(180° + θ) = tan θ. In radians the same statements hold with π in place of 180°.
相关角的事实能极大提速：sin(180° − θ) = sin θ，cos(180° − θ) = −cos θ，tan(180° − θ) = −tan θ；sin(180° + θ) = −sin θ，cos(180° + θ) = −cos θ，tan(180° + θ) = tan θ。在弧度制下把 180° 换成 π，这些结论同样成立。
7. Graphs of Trigonometric Functions | 三角函数的图像
ph of y = cos θ is the same shape shifted left by 90°; it starts at its maximum 1 when θ = 0°.
y = sin θ 的图像周期为 360°（2π），振幅为 1，且经过原点。它在 90° 处取最大值 1，在 270° 处取最小值 −1。y = cos θ 的图像只是把正弦图像向左平移 90°；当 θ = 0° 时从最大值 1 开始。
t 90° + 180°n. Remember that tan θ = sin θ / cos θ, so it is undefined wherever cos θ = 0.
y = tan θ 的图像周期为 180°（π），在 90°、270° 处有竖直渐近线，更一般地是在 90° + 180°n 处。记住 tan θ = sin θ / cos θ，因此凡 cos θ = 0 处它都无定义。
transformed function of the form y = a sin(bθ + c) + d, the key readings are: amplitude = |a| able to state these four quantities quickly is often worth two or three marks on its own.
对于形如 y = a sin(bθ + c) + d 的变换后函数，关键读数为：振幅 = |a|，周期 = 360° / b（弧度制下为 2π / b），水平平移 = −c / b，竖直平移 = d。能迅速写出这四项，本身常常就值 2 到 3 分。
Sketching is an examinable skill. Label the coordinates of maxima, minima and axis intercepts, mark the period clearly, and draw asymptotes as dashed lines for tangent graphs. A sketch without labelled key points may earn limited credit.
作图本身就是可考查的技能。要标出最大值、最小值和坐标轴交点的坐标，清楚标出周期，正切图像的渐近线用虚线画出。没有标注关键点的草图可能只能拿到有限的分数。
8. Identities You Must Know | 必须掌握的恒等式
Two identities underpin almost every manipulation in this topic:
sin² θ + cos² θ ≡ 1 and tan θ ≡ sin θ / cos θ
有两个恒等式支撑着本主题中几乎所有的变形运算，即上式。
The first identity can be rearranged into sin² θ ≡ 1 − cos² θ or cos² θ ≡ 1 − sin² θ. This is the standard route into equations that mix squared terms with linear terms, and into proofs where one side must be shown to equal the other.
第一个恒等式可变形为 sin² θ ≡ 1 − cos² θ 或 cos² θ ≡ 1 − sin² θ。这是处理同时含平方项与一次项的方程的标准路径，也是证明一边等于另一边的常用方法。
In Year 2 you also need the double-angle formulae:
sin 2θ ≡ 2 sin θ cos θ
cos 2θ ≡ cos² θ − sin² θ ≡ 1 − 2 sin² θ ≡ 2 cos² θ − 1
tan 2θ ≡ 2 tan θ / (1 − tan² θ)
第二年还需要掌握二倍角公式，即上面三式。
The three versions of cos 2θ are not interchangeable decoration: choosing the right version is the whole trick. If an expression contains only sines, use 1 − 2 sin² θ. If it contains only cosines, use 2 cos² θ − 1. If it contains both, use cos² θ − sin² θ.
cos 2θ 的三个版本不是可有可无的装饰：选对版本正是解题的关键。若表达式中只含正弦，用 1 − 2 sin² θ；若只含余弦，用 2 cos² θ − 1；若两者都有，用 cos² θ − sin² θ。
For proving identities, always start from the more complicated side and simplify towards the simpler one. Convert everything into sine and cosine if you get stuck, combine fractions over a common denominator, and never move terms across the identity sign as if it were an equation.
证明恒等式时，一定要从较复杂的一边出发，向较简单的一边化简。卡住时把所有函数都化成正弦和余弦，把分式通分，并且绝不要把项像解方程那样在恒等号两边移来移去。
9. Solving Basic Trigonometric Equations
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