📚 IGCSE Trigonometry Revision Guide | IGCSE 三角学复习指南
This revision guide covers every key trigonometry topic you need for the IGCSE Mathematics examination. From the fundamentals of right-angled triangles to the sine rule, cosine rule, and trigonometric graphs, each topic is explained with clear formulas, worked examples, and common pitfalls to avoid.
本复习指南涵盖 IGCSE 数学考试所需的全部核心三角学知识点。从直角三角形的基础知识到正弦定理、余弦定理和三角函数图像，每个主题都配有清晰的公式、典型例题以及需要避免的常见错误。
1. Right-Angled Triangles: SOH CAH TOA | 直角三角形：SOH CAH TOA
ne = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.
对于任意直角三角形，三个基本三角比将角度与边长联系起来。助记口诀 SOH CAH TOA 帮助你记住使用哪个比值：正弦 = 对边 ÷ 斜边，余弦 = 邻边 ÷ 斜边，正切 = 对边 ÷ 邻边。
tenuse cos θ = Adjacent / Hypotenuse tan θ = Opposite / Adjacent
identify the hypotenuse, opposite, and adjacent sides relative to the given angle before substituting into the formula.
考虑一个斜边为 13 cm 且含 30° 角的直角三角形。求 30° 角的对边时，使用 sin 30° = 对边 / 13，得到 对边 = 13 × sin 30° = 6.5 cm。在代入公式前，务必先相对于已知角标出斜边、对边和邻边。
Hypotenuse: the longest side, opposite the right angle.
斜边：最长的边，位于直角对面。
Opposite: the side directly across from the angle θ.
对边：与角 θ 正相对的边。
Adjacent: the side next to angle θ that is not the hypotenuse.
邻边：与角 θ 相邻但非斜边的边。
2. Exact Values of Special Angles | 特殊角的精确值
You must memorise the exact trigonometric values for the special angles 0°, 30°, 45°, 60°, and 90°. These values appear frequently in non-calculator papers and allow you to solve problems without rounding errors.
你必须熟记特殊角 0°、30°、45°、60° 和 90° 的精确三角值。这些值在非计算器试卷中频繁出现，可帮助你避免四舍五入误差。
| Angle / 角度 θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
A quick way to recall these is to note the pattern: sin goes 0, ½, √2/2, √3/2, 1 as the angle increases from 0° to 90°, while cos goes in reverse order. The tan column is simply sin divided by cos.
快速记忆方法：当角度从 0° 增加到 90° 时，sin 依次为 0、½、√2/2、√3/2、1，而 cos 则恰好相反。tan 列就是 sin 除以 cos 的结果。
3. The Sine Rule | 正弦定理
The sine rule relates the sides of any triangle to the sines of their opposite angles. It is used when you know two angles and one side (AAS) or two sides and a non-included angle (SSA).
正弦定理将任意三角形的边与其对角的正弦联系起来。它适用于已知两角一边（AAS）或两边及其中一边的对角（SSA）的情形。
a / sin A = b / sin B = c / sin C
40°, and B = 65°, then b / sin 65° = 8 / sin 40°, so b = 8 × sin 65° ÷ sin 40° ≈ 11.3 cm.
其中，边 a 对着角 A，边 b 对着角 B，边 c 对着角 C。实际解题时通常只使用其中两对关系。例如，若 a = 8 cm，A = 40°，B = 65°，则 b / sin 65° = 8 / sin 40°，因此 b = 8 × sin 65° ÷ sin 40° ≈ 11.3 cm。
To find a missing side: put the unknown side in the numerator.
求未知边：将未知边放在分子位置。
To find a missing angle: it is often easier to invert the rule.
求未知角：通常将等式倒过来更方便。
When finding an angle with SSA, beware of the ambiguous case: two possible angles may exist.
用 SSA 求角时，小心"模糊情形"：可能存在两个符合条件的角。
4. The Cosine Rule | 余弦定理
The cosine rule is used for triangles where you know two sides and the included angle (SAS), or all three sides (SSS). It extends Pythagoras' theorem to non-right-angled triangles.
余弦定理适用于已知两边及其夹角（SAS）或已知三边（SSS）的三角形。它将勾股定理推广到非直角三角形。
a² = b² + c² − 2bc × cos A
= 12, then cos A = (81 + 144 − 49) ÷ (2 × 9 × 12) = 176 ÷ 216 ≈ 0.8148, giving A ≈ 35.4°.
当已知三边求角时，可将公式变形为：cos A = (b² + c² − a²) ÷ (2bc)。例如，若 a = 7，b = 9，c = 12，则 cos A = (81 + 144 − 49) ÷ (2 × 9 × 12) = 176 ÷ 216 ≈ 0.8148，得 A ≈ 35.4°。
90° = 0, and the rule correctly reduces to a² = b² + c², your familiar Pythagoras theorem.
注意：待求的边 a 始终对着公式中的角 A。当角为 90° 时，cos 90° = 0，余弦定理自动退化为 a² = b² + c²，即你熟悉的勾股定理。
5. Area of a Triangle | 三角形面积公式
For any triangle, the area can be found from two sides and the included angle without needing the perpendicular height. This formula is part of the IGCSE core syllabus.
对于任意三角形，已知两边及其夹角即可求面积，无需知道高。该公式属于 IGCSE 核心考纲内容。
Area = ½ × a × b × sin C
cm and 14 cm with an included angle of 50°, the area is ½ × 10 × 14 × sin 50° ≈ 53.6 cm².
其中，边 a 和边 b 是已知的两条边，角 C 是它们的夹角。例如，若两边分别为 10 cm 和 14 cm，夹角为 50°，则面积为 ½ × 10 × 14 × sin 50° ≈ 53.6 cm²。
Remember that the included angle is always between the two chosen sides.
记住：夹角必是所选两边的夹角。
perpendicular height is given instead, the simpler Area = ½ × base × height still applies.
若题目给出的是高，则仍可使用更简单的 面积 = ½ × 底 × 高。
Always include square units (cm², m²) in your final answer.
最终答案务必带上面积单位（cm²、m²）。
6. Graphs of Trigonometric Functions | 三角函数图像
an x have distinctive shapes that you must recognise and sketch. They are periodic, meaning they repeat at regular intervals.
y = sin x、y = cos x 和 y = tan x 的图像各具特征，你必须能够识别和绘制。它们都是周期函数，图像按固定间隔重复。
in x: passes through the origin, rises to a maximum of 1 at 90°, and has a period of 360°.
y = sin x：过原点，在 90° 处达到最大值 1，周期为 360°。
y = cos x: starts at a maximum of 1 at 0°, and also has a period of 360°.
y = cos x：在 0° 处从最大值 1 开始，周期同样为 360°。
tan x: passes through the origin, has asymptotes at 90° and 270°, and repeats every 180°.
y = tan x：过原点，在 90° 和 270° 处有渐近线，周期为 180°。
Be aware of transformations: y = a sin x changes the amplitude to |a| riod to 360°/b. A negative sign, such as y = −cos x, reflects the graph across the x-axis.
注意图像变换：y = a sin x 将振幅变为 |a|，而 y = sin bx 将周期变为 360°/b。负号如 y = −cos x 表示将图像沿 x 轴翻转。
7. Key Trigonometric Identities | 重要三角恒等式
Two fundamental identities are essential for simplifying expressions and solving equations. They are derived from the definitions of sin, cos, and tan on the unit circle.
两个基本恒等式对于化简表达式和解方程至关重要。它们由单位圆上 sin、cos 和 tan 的定义推导得出。
tan θ = sin θ / cos θ sin²θ + cos²θ = 1
For example, if sin θ = 3/5 and θ is acute, then cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5.
第一个恒等式可将含正切的表达式改写。第二个恒等式称为毕达哥拉斯恒等式，特别适合在已知一个三角比值时求另一个。例如，若 sin θ = 3/5 且 θ 为锐角，则 cos²θ = 1 − 9/25 = 16/25，故 cos θ = 4/5。
8. Solving Trigonometric Equations | 解三角方程
Solving trig equations on a given interval usually produces multiple solutions. The number of solutions depends on the period of the function and the range specified in the question.
在给定区间内解三角方程通常会得到多个解。解的个数取决于函数的周期以及题目指定的范围。
Example: Solve 2 sin x = 1 for 0° ≤ x ≤ 360°
nd second quadrants, the second solution is x = 180° − 30° = 150°. So x = 30° or x = 150°.
首先除以 2 得到 sin x = 0.5。主解为 x = 30°。由于正弦在第一、二象限均为正，第二个解为 x = 180° − 30° = 150°。因此 x = 30° 或 x = 150°。
Always find the principal value using your calculator first.
先用计算器求出主值。
Use the CAST diagram or the graph to identify all quadrants where the ratio is positive or negative.
利用 CAST 象限图或函数图像判断该比值为正或负的所有象限。
Check that every solution lies inside the specified interval.
检查所有解是否都在指定区间内。
9. Bearings and Angles of Elevation | 方位角与仰角俯角
Trigonometry is widely applied to problems involving bearings, angles of elevation, and angles of depression. Bearings are measured clockwise from north and are always written as three-digit angles.
三角学广泛用于方位角、仰角和俯角问题。方位角从正北方向顺时针测量，且始终写成三位数形式。
For an angle of elevation, look from the horizontal up to an object above; for an angle of depression, look from the horizontal down to an object below. In both cases, the right-angled triangle is set up with the horizontal as one leg and the vertical difference as the other leg.
仰角是从水平线向上看物体的角度；俯角是从水平线向下看物体的角度。两种情况都构成直角三角形，水平距离为一条直角边，垂直高度差为另一条直角边。
Height = Horizontal distance × tan (angle of elevation)
A classic problem: a tower casts a shadow 25 m long when the sun' of the tower is 25 × tan 35° ≈ 17.5 m. Remember to round your final answer appropriately.
经典问题：当太阳仰角为 35° 时，一座塔的影子长为 25 m。塔高为 25 × tan 35° ≈ 17.5 m。记得合理取舍最终答案。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many marks are lost on trigonometry questions not because of difficult mathematics, but because of small, avoidable errors. Here are the most common pitfalls and how to avoid them.
三角学题目失分往往不是因为数学本身困难，而是因为一些细小且可避免的错误。以下是最常见的陷阱及规避方法。
| Mistake / 常见错误 | Correction / 纠正方法 |
|---|---|
| Calculator in radian mode | Always check the DEG mode indicator before starting. |
| 计算器处于弧度模式 | 开始前务必检查屏幕上的 DEG 度数模式标志。 |
| Swapping opposite and adjacent sides | Label the triangle clearly relative to the angle in use. |
| 将"对边"与"邻边"混淆 | 先相对所用角清晰地标出各边。 |
| Forgetting the ambiguous case in the sine rule | When given SSA, check if 180° − principal angle is also valid. |
| 正弦定理中遗漏"模糊情形" | 已知 SSA 时，检查 180° − 主角是否同样成立。 |
| Rounding prematurely | Keep full precision until the final step, then round to 3 s.f. |
| 过早四舍五入 | 保留完整精度直到最后一步，再四舍五入到 3 位有效数字。 |
In the exam, sketch a clear diagram for every problem, even if one is not given. Write down the formula you are using before substituting numbers, and ensure your final answer has the correct units. Practise past-paper questions until the SOH CAH TOA, sine rule, and cosine rule become second nature.
考试中，即使题目没有给出图，也要为每道题画一个清晰的示意图。在代入数字前先写出所用公式，并确保最终答案带有正确单位。反复练习历年真题，直到 SOH CAH TOA、正弦定理和余弦定理成为你的本能反应。
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