📚 Chapter Review 9: Trigonometric Ratios | 第9章复习：三角比
This chapter review covers the core ideas of trigonometric ratios for Edexcel A-Level Mathematics. You will revisit the definitions of sine, cosine and tangent, exact values for key angles, radians, the unit circle, graphs, the two fundamental identities, and strategies for solving simple trigonometric equations. Mastery of these basics is essential before moving on to trigonometric identities and equations in later chapters.
本章复习涵盖 Edexcel A-Level 数学中三角比的核心概念。你将重温正弦、余弦和正切的定义，关键角的精确值、弧度、单位圆、函数图像、两个基本恒等式，以及解简单三角方程的策略。在进入后续章节的三角恒等式与方程之前，牢固掌握这些基础至关重要。
1. The Sine, Cosine and Tangent Ratios | 正弦、余弦和正切比
For a right-angled triangle, label the sides relative to an acute angle θ. The hypotenuse is the longest side and is always opposite the right angle. The opposite side is across from θ, and the adjacent side is the remaining side, next to θ but not the hypotenuse.
对于直角三角形，相对于锐角 θ 标记边。斜边是最长边，永远对着直角。对边在 θ 的对面，邻边是剩下那条紧邻 θ 但不是斜边的边。
The three basic trigonometric ratios are defined as follows, where opposite means the side opposite θ, adjacent means the side adjacent to θ, and hypotenuse is the longest side.
三个基本三角比定义如下，其中 opposite 表示 θ 的对边，adjacent 表示 θ 的邻边，hypotenuse 是最长边。
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
These ratios only apply to right-angled triangles. For non-right-angled triangles, you will need the sine rule or cosine rule, which are covered in later chapters.
这些比只适用于直角三角形。对于非直角三角形，你需要使用正弦定理或余弦定理，它们将在后续章节中学习。
2. SOH CAH TOA and Labelling Conventions | SOH CAH TOA 与标注约定
Remember the abbreviation SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. This mnemonic should be automatic before you attempt any exam question.
记住缩写 SOH CAH TOA：正弦是对边除以斜边，余弦是邻边除以斜边，正切是对边除以邻边。在做任何考题之前，这个口诀都应该成为条件反射。
tion: θ = sin⁻¹(opposite / hypotenuse), θ = cos⁻¹(adjacent / hypotenuse) or θ = tan⁻¹(opposite / adjacent).
要求未知边，就把两个已知值代入正确的比。要求未知角，就重排该比并套用反函数：θ = sin⁻¹(对边 / 斜边)、θ = cos⁻¹(邻边 / 斜边) 或 θ = tan⁻¹(对边 / 邻边)。
A common trap is choosing the wrong ratio. First write down which two sides are involved, then decide which of the three ratios links exactly those two sides.
常见的陷阱是选错比。先写下涉及的是哪两条边，再判断三个比中哪一个正好连接这两条边。
3. Exact Values for Special Angles | 特殊角的精确值
You must know the exact values for 0°, 30°, 45°, 60° and 90° without a calculator. These arise from an equilateral triangle of side 2 (for 30° and 60°) and an isosceles right triangle with legs 1 (for 45°).
你必须在不使用计算器的情况下知道 0°、30°、45°、60° 和 90° 的精确值。它们分别来自边长为 2 的等边三角形（给出 30° 和 60°）以及直角边为 1 的等腰直角三角形（给出 45°）。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | 1/2 | √2 / 2 | √3 / 2 | 1 |
| cos θ | 1 | √3 / 2 | √2 / 2 | 1/2 | 0 |
| tan θ | 0 | √3 / 3 | 1 | √3 | 未定义 / undefined |
Notice the pattern: the sine values increase from 0 to 1, the cosine values decrease from 1 to 0, and the tangent values increase without bound as θ approaches 90°. Answers should be given in exact surd form unless the question states otherwise.
注意其中的规律：正弦值从 0 增至 1，余弦值从 1 减至 0，而正切值在 θ 趋近 90° 时无限增大。除非题目另有说明，答案应以精确根式形式给出。
4. Radians and Degrees | 弧度与角度
based on the fact that a full turn is 360°, which equals 2π radians, so 180° = π radians.
角度可以用度或弧度来度量。换算依据是一个整圈为 360°，等于 2π 弧度，因此 180° = π 弧度。
angle in radians = angle in degrees × π / 180°, angle in degrees = angle in radians × 180° / π
= ½ r² θ always require radians, so the exact value table should be learnt in both forms.
相应的特殊角变成 π/6、π/4、π/3 和 π/2。涉及微积分、弧长 s = rθ 和扇形面积 A = ½ r² θ 的 A-Level 题目一律要求弧度，因此精确值表应以两种形式同时掌握。
Before starting any calculation, check that your calculator is in the correct mode, DEG or RAD. A wrong mode is one of the most expensive mistakes in trigonometry papers.
开始任何计算之前，请确认计算器处于正确模式（DEG 或 RAD）。模式选错是三角题中最"贵"的错误之一。
5. The Unit Circle and General Angles | 单位圆与任意角
The unit circle is a circle of radius 1 centred at the origin. If a point P on the circle is reached by rotating an angle θ anticlockwise from the positive x-axis, then the coordinates of P are (cos θ, sin θ).
单位圆是以原点为圆心、半径为 1 的圆。若从 x 轴正方向逆时针旋转角 θ 到达圆上一点 P，则 P 的坐标就是 (cos θ, sin θ)。
Since the radius is 1, the definitions of the ratios extend naturally to any angle, obtuse, reflex or negative. The x-coordinate gives the cosine and the y-coordinate gives the sine, so both lie between −1 and 1.
因为半径是 1，三角比的定义自然可以推广到任意角——钝角、大于 180° 的角甚至负角。横坐标给出余弦，纵坐标给出正弦，因此两者都在 −1 与 1 之间。
tan θ = sin θ / cos θ = y / x, which is undefined whenever x = 0, that is at 90° and 270°.
正切也可以由同一幅图得到：tan θ = sin θ / cos θ = y / x，当 x = 0 即 90° 和 270° 时无定义。
6. Signs in the Four Quadrants | 四个象限中的符号
Because cos θ is the x-coordinate and sin θ is the y-coordinate, the signs of the three ratios change from quadrant to quadrant. The standard mnemonic is CAST, read anticlockwise from the fourth quadrant: All, Sine, Tangent, Cosine.
由于 cos θ 是横坐标、sin θ 是纵坐标，三个比的符号会随象限改变。标准口诀是 CAST，从第四象限起逆时针读作：All（全正）、Sine（正弦为正）、Tangent（正切为正）、Cosine（余弦为正）。
- First quadrant (0° to 90°): all three ratios are positive. | 第一象限（0° 到 90°）：三个比全为正。
- Second quadrant (90° to 180°): only sine is positive. | 第二象限（90° 到 180°）：只有正弦为正。
- Third quadrant (180° to 270°): only tangent is positive. | 第三象限（180° 到 270°）：只有正切为正。
- Fourth quadrant (270° to 360°): only cosine is positive. | 第四象限（270° 到 360°）：只有余弦为正。
The related acute angle, or reference angle, is the acute angle between the radius and the x-axis. Every solution of a trigonometric equation can be generated from a reference angle together with the correct quadrant sign.
相关的锐角（参考角）是半径与 x 轴之间的锐角。三角方程的每一个解都可以由参考角加上正确的象限符号生成。
7. Graphs of the Three Functions | 三个函数的图像
2π. The graph of y = cos x has the same shape but starts at its maximum, since cos 0° = 1.
y = sin x 的图像是一条过原点的波浪曲线，在 1 与 −1 之间振动，周期为 360° 即 2π。y = cos x 的图像形状相同，但从最大值开始，因为 cos 0° = 1。
t: it has period 180° or π, repeats its shape with vertical asymptotes at 90° and 270°, and has no maximum or minimum value.
y = tan x 的图像则很不相同：周期为 180° 即 π，在 90° 和 270° 处有竖直渐近线，并且没有最大值或最小值。
Sketching the graph is often the quickest way to see how many solutions an equation has in a given interval, especially when the question asks for solutions between 0° and 360°.
画出草图通常是判断方程在给定区间内有多少个解的最快方法，尤其当题目要求 0° 到 360° 之间的解时。
8. Two Fundamental Identities | 两个基本恒等式
Two identities follow directly from the unit circle and must be memorised. The first is the Pythagorean identity, obtained by applying Pythagoras' theorem to the coordinates (cos θ, sin θ).
有两个恒等式直接来自单位圆，必须牢记。第一个是毕达哥拉斯恒等式，由对坐标 (cos θ, sin θ) 使用勾股定理得到。
sin² θ + cos² θ = 1
The second relates tangent to sine and cosine. It is valid whenever cos θ ≠ 0, and it is the reason tan θ is undefined at 90° and 270°.
第二个恒等式把正切与正弦、余弦联系起来。它在 cos θ ≠ 0 时成立，也是 tan θ 在 90° 和 270° 处无定义的原因。
tan θ = sin θ / cos θ
These two identities let you simplify expressions, prove simple results, and find sin θ from cos θ (or the reverse) once the quadrant is known to fix the sign.
这两个恒等式可用于化简表达式、证明简单结论，以及在已知象限确定符号后由 cos θ 求 sin θ（或反过来）。
9. Solving Simple Trigonometric Equations | 解简单三角方程
mmetry of the graph, or the CAST diagram, to find the remaining solutions in the interval.
解 sin x = 0.5 且 0° ≤ x ≤ 360° 这类方程时，先用反函数求主值：x = sin⁻¹ 0.5 = 30°。再利用图像对称性或 CAST 图找出区间内的其余解。
- For sin x = k: the two solutions in one full turn are θ and 180° − θ. | 对 sin x = k：一整圈内的两个解是 θ 和 180° − θ。
- For cos x = k: the two solutions are θ and 360° − θ. | 对 cos x = k：两个解是 θ 和 360° − θ。
- For tan x = k: the two solutions are θ and θ + 180°. | 对 tan x = k：两个解是 θ 和 θ + 180°。
use the tangent has period 180°. If the interval is wider, keep adding or subtracting the period until you leave the range.
例如，tan x = 1 在 0° 到 360° 内给出 x = 45° 和 x = 225°，因为正切周期为 180°。若区间更宽，就反复加或减周期，直到超出范围为止。
10. Right-Angled Triangle Problems in Context | 情境中的直角三角形问题
Many exam questions place the triangle inside a real situation: a ladder against a wall, the height of a tree, or the angle of elevation from a point on the ground to the top of a building.
许多考题把三角形放在真实情境中：梯子靠墙、树的高度，或从地面某点到建筑物顶部的仰角。
The angle of elevation is measured upwards from the horizontal, while the angle of depression is measured downwards from the horizontal. These two angles are equal when taken as alternate angles between two parallel horizontal lines.
仰角是从水平线向上量的角，俯角是从水平线向下量的角。当两条水平线平行时，这两个角作为内错角相等。
Draw a clear diagram, mark the right angle, and label every known side and angle before choosing a ratio. If the triangle is not right-angled, SOH CAH TOA does not apply and you must switch to the sine rule or cosine rule.
画清示意图，标出直角，并在选择比之前标出所有已知边和角。若三角形不是直角三角形，SOH CAH TOA 不适用，必须改用正弦定理或余弦定理。
11. Common Errors and Exam Technique | 常见错误与考试技巧
Several mistakes appear again and again in trigonometry. Being aware of them is often worth more marks than learning new content.
三角学中反复出现几类错误。意识到它们，往往比学新知识更能提分。
- Calculator in the wrong mode (DEG instead of RAD, or the reverse). | 计算器模式错误（该用 RAD 却用 DEG，或相反）。
- Using SOH CAH TOA on a triangle that has no right angle. | 在非直角三角形上使用 SOH CAH TOA。
- Giving only the principal value when the interval demands more solutions. | 区间内需要多个解时却只给出主值。
- Losing the negative sign when the angle lies in the second or third quadrant. | 角位于第二或第三象限时丢失负号。
- Rounding intermediate values too early, which distorts the final answer. | 过早对中间结果取近似，导致最终答案失真。
Always state the formula you are using, keep full accuracy in your calculator, and round only at the end. Where the question says "hence" or "show that", write out the logical steps rather than jumping to the answer.
务必写出你使用的公式，计算器内保留完整精度，只在最后一步取近似。当题目出现 "hence" 或 "show that" 时，要写出逻辑步骤，而不是直接跳到答案。
12. Chapter Summary Checklist | 章节自检清单
Before moving on, check that you can do each of the following without notes. This list mirrors the skills that later chapters on identities and equations will assume.
在进入下一章之前，请确认以下每一项都能脱离笔记完成。这份清单正对应后续恒等式与方程章节所假定的技能。
- State the three ratios and recall SOH CAH TOA. | 说出三个比并回忆 SOH CAH TOA。
- Write down the exact values for 0°, 30°, 45°, 60° and 90°. | 写出 0°、30°、45°、60°、90° 的精确值。
- Convert between degrees and radians fluently. | 熟练在度与弧度之间换算。
- Use the unit circle to define sin θ and cos θ for any angle. | 用单位圆定义任意角的 sin θ 和 cos θ。
- Apply the CAST diagram to determine signs. | 用 CAST 图判断符号。
- Sketch y = sin x, y = cos x and y = tan x with correct periods. | 正确画出 y = sin x、y = cos x、y = tan x 并标出周期。
- Quote and use sin² θ + cos² θ = 1 and tan θ = sin θ / cos θ. | 引用并使用 sin² θ + cos² θ = 1 与 tan θ = sin θ / cos θ。
- Solve sin x = k, cos x = k and tan x = k over a stated interval. | 在给定区间内解 sin x = k、cos x = k、tan x = k。
- Model a real situation with a right-angled triangle. | 用直角三角形建模真实情境。
Once these basics are secure, identities, compound angles and the full theory of trigonometric equations become far more manageable, because almost every later technique reduces to the ideas reviewed in this chapter.
一旦这些基础牢固，恒等式、复合角以及完整的三角方程理论都会变得容易得多，因为后续几乎每一种技巧都可以化归到本章复习的这些思想。
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