📚 Trigonometry Essentials for IGCSE Mathematics | IGCSE数学三角函数精讲
Trigonometry is one of the most rewarding topics in the IGCSE Mathematics syllabus. It appears in both Paper 2 and Paper 4, and it connects naturally with geometry, algebra and coordinate graphs. Once you master the core ratios, the sine and cosine rules, and the graphs of trig functions, you will find that exam questions become highly predictable and easy to score on.
三角函数是IGCSE数学课程中最容易拿分、性价比最高的专题之一。它在Paper 2和Paper 4中均有出现，并与几何、代数和坐标图像紧密相连。一旦你掌握了基本三角比、正弦定理、余弦定理以及三角函数图像，你会发现考试中的相关题目套路清晰、得分稳定。
1. Understanding Right-Angled Triangles | 理解直角三角形
Every trigonometric ratio in the IGCSE course begins with a right-angled triangle. A right-angled triangle contains one 90° angle, and the longest side is always called the hypotenuse (hyp). The two shorter sides are named relative to a chosen acute angle θ: the side next to θ is the adjacent side (adj), and the side opposite θ is the opposite side (opp).
IGCSE课程中的每一个三角比都从直角三角形开始。直角三角形有一个90°角，最长的边始终称为斜边（hypotenuse）。另外两条较短的边是相对于选定的锐角θ来命名的：紧挨着θ的边称为邻边（adjacent），与θ相对的边称为对边（opposite）。
| Side | Position in triangle |
| Hypotenuse (hyp) | Longest side, opposite the right angle |
| Opposite (opp) | Directly across from angle θ |
| Adjacent (adj) | Next to angle θ, not the hypotenuse |
Always label the sides on your diagram before you write any equation. This single habit prevents most careless errors in trigonometry questions.
在写任何方程之前，一定要先在图上标出三条边的名称。这一个好习惯能避免三角函数题中大部分粗心错误。
2. The Three Basic Ratios | 三大基本三角比
The three primary trigonometric ratios are sine, cosine and tangent. They are commonly remembered by the mnemonic SOH CAH TOA, which stands for:
三大基本三角比是正弦（sine）、余弦（cosine）和正切（tangent）。通常用口诀SOH CAH TOA来记忆，即：
sin θ = opp/hyp cos θ = adj/hyp tan θ = opp/adj
Each ratio is a fraction that compares two side lengths of a right-angled triangle. The value of the ratio depends only on the angle θ, not on the size of the triangle. That is why your calculator returns the same sine value for 30° whether the triangle is small or large.
每个三角比都是一个比较直角三角形两条边长的分数。比值只取决于角度θ，而与三角形的大小无关。因此，无论三角形是大是小，计算器给出的30°正弦值都是相同的。
You must decide which triangle side is opposite and which is adjacent first. Once the sides are labelled correctly, choosing the correct ratio becomes a straightforward matching exercise.
你首先要判断哪条边是对边、哪条边是邻边。一旦边长标签标对，选择正确的三角比就变成了一项简单的匹配工作。
3. Finding Missing Sides | 求未知边长
To find a missing side in a right-angled triangle, follow three steps. First, label the hypotenuse, opposite and adjacent sides relative to the given angle. Second, select the ratio that contains both the known side and the unknown side. Third, rearrange the equation so the unknown side becomes the subject, then use your calculator.
求直角三角形中的未知边长，只需三步。第一步：相对于已知角标出斜边、对边和邻边。第二步：选择同时包含已知边和未知边的那一个三角比。第三步：移项使未知边成为公式主项，再用计算器求值。
Example: A ladder of length 6 m leans against a wall, making an angle of 62° with the ground. Find the height reached by the ladder.
例题：一架长6米的梯子斜靠在墙上，与地面成62°角，求梯子顶端到达的高度。
The height h is opposite the 62° angle, and the ladder is the hypotenuse. Therefore sin 62° = h/6.
高度h是62°角的对边，梯子是斜边，因此sin 62° = h/6。
h = 6 × sin 62° ≈ 5.30 m
Always write the full equation before substituting the numbers. This shows the examiner your method and earns method marks even if your final calculation is wrong.
一定要先写出完整方程再代入数值。这样既能向考官展示你的思路，即使最后算错也能获得方法分。
4. Finding Missing Angles | 求未知角度
When two side lengths are known and you need to find an angle, you use the inverse trigonometric functions: sin⁻¹, cos⁻¹ and tan⁻¹. On most calculators these are the second function above the sin, cos and tan buttons.
当已知两条边长而需要求角度时，就要使用反三角函数：sin⁻¹、cos⁻¹和tan⁻¹。大多数计算器上，这些功能位于sin、cos和tan按键的第二功能层。
θ = sin⁻¹(opp/hyp) θ = cos⁻¹(adj/hyp) θ = tan⁻¹(opp/adj)
Example: In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 12 cm. Find θ.
例题：直角三角形中对边长为7 cm，斜边长为12 cm，求角θ。
θ = sin⁻¹(7/12) ≈ 35.7°
Make sure your calculator is in degree mode, not radian mode. A wrong mode is the most common cause of incorrect angle answers in IGCSE exams.
请确保计算器处于角度制（degree mode）而不是弧度制（radian mode）。角度模式错误是IGCSE考试中最常见的丢分原因。
5. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal line of sight to an object above the observer. An angle of depression is the angle measured downwards from the horizontal line of sight to an object below the observer.
仰角是从水平视线向上测量到观察者上方物体的角度；俯角是从水平视线向下测量到观察者下方物体的角度。
When solving these problems, always draw a clear diagram first. Mark the horizontal line, the right angle and the given angle. The vertical height and the horizontal distance become the opposite and adjacent sides of a right-angled triangle, so you can apply the tangent ratio directly.
解这类问题时，务必先画清楚示意图。标出水平线、直角和已知角。竖直高度与水平距离构成直角三角形的对边和邻边，因此可以直接使用正切比。
Remember that the angle of elevation from point A to point B equals the angle of depression from point B to point A, because the two horizontal lines are parallel. This symmetry is frequently tested in IGCSE questions.
请记住，从A点观察B点的仰角等于从B点观察A点的俯角，因为两条水平线互相平行。这一对称性质在IGCSE考题中经常出现。
6. Bearings and Trigonometry | 方位角与三角函数
In IGCSE Mathematics, a bearing is a three-figure angle measured clockwise from north. For example, a bearing of 045° means 45° clockwise from north, and a bearing of 135° means 135° clockwise from north. Bearings are always written with exactly three digits.
在IGCSE数学中，方位角是从正北方向顺时针测量的三位数角度。例如，方位角045°表示从正北顺时针旋转45°，方位角135°表示从正北顺时针旋转135°。方位角必须用恰好三位数字表示。
Trigonometry converts a journey distance and direction into north-south and east-west displacements. Suppose a ship travels a distance d on a bearing θ. Its northward displacement is given by d cos θ, and its eastward displacement is given by d sin θ.
三角函数可以将一段航行的距离和方向转换为南北方向与东西方向的位移。假设一艘船沿方位角θ航行了距离d，则其向北的位移为d cos θ，向东的位移为d sin θ。
For bearings beyond 90°, sketch the direction carefully and use the acute angle between the line of travel and the nearest compass axis. Drawing the right-angled triangle is always the key first step.
对于大于90°的方位角，请仔细画方向草图，并使用航行方向与最近的指南针轴之间的锐角。画出正确的直角三角形始终是关键第一步。
7. The Sine Rule | 正弦定理
For any triangle that is not right-angled, you need the sine rule and the cosine rule. The sine rule relates each side to the sine of its opposite angle:
对于任何非直角三角形，需要用到正弦定理和余弦定理。正弦定理将每条边与其对角的正弦联系起来：
a/sin A = b/sin B = c/sin C
Here a is the side opposite angle A, b is opposite angle B, and c is opposite angle C. Use the sine rule when you know two angles and one side (AAS), or when you know two sides and a non-included angle (SSA).
其中a是角A的对边，b是角B的对边，c是角C的对边。当已知两角和一边（AAS），或已知两边和一个非夹角（SSA）时，使用正弦定理。
Example: In triangle ABC, angle A = 35°, angle B = 70° and side a = 8 cm. Find side b.
例题：在三角形ABC中，角A = 35°，角B = 70°，边a = 8 cm，求边b。
b/sin 70° = 8/sin 35° → b = 8 × sin 70° / sin 35° ≈ 13.1 cm
When finding an angle using the sine rule, take extra care: the equation may produce two possible angles because sin θ = sin(180° − θ). Check whether the obtuse angle is geometrically possible.
用正弦定理求角时要格外小心：方程可能产生两个可能的角度，因为sin θ = sin(180° − θ)。请检查钝角在几何上是否可能成立。
8. The Cosine Rule | 余弦定理
The cosine rule is used when you know two sides and the included angle (SAS), or when you know all three sides (SSS) and wish to find an angle. The rule has two equivalent forms:
余弦定理用于已知两边及其夹角（SAS），或已知三边（SSS）求某个角的情形。它有如下两种等价形式：
a² = b² + c² − 2bc cos A
cos A = (b² + c² − a²) / (2bc)
The first form finds a missing side; the second form, obtained by rearranging, finds a missing angle. Notice that the side on the left of the first equation is always opposite the angle on the right.
第一个公式用于求未知边；第二个公式由第一个移项得到，用于求未知角。注意，第一个公式左边的边永远是右边那个角的对边。
Example: Two sides of a triangle are 7 cm and 5 cm, and the included angle is 45°. Find the third side.
例题：三角形两边分别为7 cm和5 cm，夹角为45°，求第三边。
c² = 7² + 5² − 2 × 7 × 5 × cos 45° = 74 − 35√2 ≈ 24.5 → c ≈ 4.95 cm
When applying the cosine rule, always square the side lengths before subtracting the cosine term, and round your answer only at the final step to avoid accumulation of error.
使用余弦定理时，一定要先平方边长，再减去余弦项；并且只在最后一步四舍五入，避免误差累积。
9. Area of a Triangle | 三角形面积公式
Beyond the basic formula ½ × base × height, IGCSE extended mathematics requires the sine area formula. For any triangle with two known sides a and b and the included angle C between them:
除了½ × 底 × 高这一基本面积公式外，IGCSE拓展数学还要求掌握正弦面积公式。对于任意两条已知边a、b及其夹角C：
Area = ½ ab sin C
This formula is especially useful when the perpendicular height is not easily found. It works for right-angled triangles, acute triangles and obtuse triangles alike, provided C is the angle between the two chosen sides.
这个公式在垂线高度不易求出时特别有用。只要C是所选两条边的夹角，无论直角三角形、锐角三角形还是钝角三角形均适用。
Example: Find the area of a triangle with sides 6 cm and 8 cm and an included angle of 30°.
例题：已知三角形两边为6 cm和8 cm，夹角为30°，求面积。
Area = ½ × 6 × 8 × sin 30° = 12 cm²
Be mindful that the sine of an obtuse angle is positive, so the area formula works correctly for obtuse triangles too. Do not use the cosine rule result here unless you truly need the third side first.
注意钝角的正弦值为正，因此该面积公式对钝角三角形同样正确。除非确实需要先求第三边，否则不要在此处使用余弦定理。
10. Graphs of Trigonometric Functions | 三角函数图像
The graphs of y = sin x and y = cos x are periodic waves that oscillate between −1 and 1, with a period of 360°. The graph of y = tan x has a very different shape: it repeats every 180° and has vertical asymptotes at 90° and 270°.
y = sin x和y = cos x的图像是在−1和1之间振荡的周期波形，周期为360°。y = tan x的图像形状截然不同：它每180°重复一次，并在90°和270°处有垂直渐近线。
You should be able to sketch these graphs and know their key features: maximum and minimum values, intercepts with the axes, and the positions of asymptotes. Exam questions often ask you to solve an equation graphically by reading the points of intersection.
你应该能够画出这些图像并掌握其关键特征：最大值和最小值、与坐标轴的交点以及渐近线的位置。考试题经常要求你通过读取图像交点来求解方程。
| θ | 0° | 30° | 45° | 60° | 90° |
| sin θ | 0 | ½ | √2/2 | √3/2 | 1 |
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