Trigonometry for IGCSE: Ratios, Rules and Applications | IGCSE 三角函数：比例、定理与应用

📚 Trigonometry for IGCSE: Ratios, Rules and Applications | IGCSE 三角函数：比例、定理与应用

Trigonometry is one of the most practical and exam-relevant topics in IGCSE Mathematics. It connects angles and side lengths, and appears in geometry, problem solving, and even statistics. Mastering the key ratios and rules will help you unlock marks quickly.

三角学是 IGCSE 数学中最实用、最贴近考点的主题之一。它将角度与边长联系起来，出现在几何、解题甚至统计中。掌握核心比例与定理，能帮助你快速得分。

1. What is Trigonometry? | 什么是三角学？

Trigonometry comes from Greek words meaning “triangle measurement”. It studies the relationships between the angles and sides of triangles, starting with right-angled triangles and extending to any triangle.

三角学源自希腊语，意为“三角形测量”。它研究三角形中角度与边之间的关系，从直角三角形开始，并推广到任意三角形。

In IGCSE, you will use trigonometric ratios (sin, cos, tan), the sine rule, the cosine rule, and the area formula. You will also interpret graphs of trigonometric functions and solve basic equations.

在 IGCSE 中，你需要使用三角比（sin、cos、tan）、正弦定理、余弦定理和面积公式，还要理解三角函数图像并解基本三角方程。

2. Basic Trigonometric Ratios | 基本三角比

For a right-angled triangle with an angle θ, the three main ratios are defined using the sides: opposite (O), adjacent (A), and hypotenuse (H). The hypotenuse is always the longest side.

在直角三角形中，对于角度 θ，三个主要比例用对边（O）、邻边（A）和斜边（H）定义。斜边始终是最长的边。

sin θ = O / H cos θ = A / H tan θ = O / A

A common mnemonic is SOH CAH TOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.

常用记忆法 SOH CAH TOA：正弦 = 对边 ÷ 斜边，余弦 = 邻边 ÷ 斜边，正切 = 对边 ÷ 邻边。

You only use these ratios in right-angled triangles. For non-right triangles, apply the sine rule or cosine rule later in this article.

这些比例只在直角三角形中使用。对于非直角三角形，请使用后面介绍的正弦定理或余弦定理。

3. Finding Sides and Angles | 求边与角

To find an unknown side, choose the ratio that involves the known angle, the unknown side, and one other known side. Then substitute and solve.

要求未知边，应选择包含已知角、未知边和另一条已知边的三角比，然后代入求解。

Example: In a right triangle, sin 30° = x / 10. Therefore x = 10 × sin 30° = 5.

例如：在直角三角形中，sin 30° = x / 10，因此 x = 10 × sin 30° = 5。

To find an angle, use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹. For example, if tan θ = 1.5, then θ = tan⁻¹(1.5), which is about 56.3°.

求角度时使用反函数：sin⁻¹、cos⁻¹、tan⁻¹。例如，若 tan θ = 1.5，则 θ = tan⁻¹(1.5)，约为 56.3°。

Always make sure your calculator is in degree mode when angles are in degrees.

角度以度为单位时，请确保计算器处于角度（DEG）模式。

4. The Sine Rule | 正弦定理

The sine rule works for any triangle. It is useful when you know two angles and one side (AAS) or two sides and a non-included angle (SSA).

正弦定理适用于任意三角形。当你已知两角一边（AAS）或两边及其中一边的对角（SSA）时，使用它非常方便。

a / sin A = b / sin B = c / sin C

Here a, b, c are the sides opposite angles A, B, C respectively. Use the first two parts to solve for missing values.

其中 a、b、c 分别是角 A、B、C 的对边。利用前两部分即可求解缺失量。

Example: If A = 40°, B = 65°, and a = 8 cm, then b = a × sin B / sin A = 8 × sin 65° / sin 40° ≈ 11.3 cm.

例：若 A = 40°，B = 65°，a = 8 cm，则 b = a × sin B / sin A = 8 × sin 65° / sin 40° ≈ 11.3 cm。

Be careful with the ambiguous case in SSA: you may get two possible triangles. IGCSE usually avoids this, but a diagram can help you decide.

注意 SSA 可能存在“多解”情况：可能出现两个不同三角形。IGCSE 通常回避这一点，但画图可以帮助判断。

5. The Cosine Rule | 余弦定理

The cosine rule is used when you know two sides and the included angle (SAS) or three sides (SSS).

余弦定理适用于已知两边及其夹角（SAS）或三边（SSS）的情形。

a² = b² + c² − 2bc cos A

Rearranged to find an angle:

变形后可用于求角度：

cos A = (b² + c² − a²) / (2bc)

Example: If b = 5, c = 7, and A = 60°, then a² = 25 + 49 − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39. So a = √39 ≈ 6.24.

例：若 b = 5，c = 7，A = 60°，则 a² = 25 + 49 − 2 × 5 × 7 × cos 60° = 25 + 49 − 35 = 39。因此 a = √39 ≈ 6.24。

The cosine rule is more general than the sine rule. It is also a version of Pythagoras with a “correction term”.

余弦定理比正弦定理更通用，它可看作勾股定理再加上一个“修正项”。

6. Area of a Triangle | 三角形面积

The standard formula Area = ½ × base × height requires a perpendicular height. Trigonometry gives a direct formula when you know two sides and the included angle.

通常的面积公式面积 = ½ × 底 × 高需要垂直高。而三角公式在已知两边及其夹角时可直接计算。

Area = ½ ab sin C

Here a and b are two sides, and C is the angle between them. This formula works for any triangle.

其中 a 和 b 是两条边，C 是它们的夹角。该公式适用于任意三角形。

Example: A triangle has sides 6 cm and 8 cm with an included angle of 45°. Area = ½ × 6 × 8 × sin 45° = 24 × √2/2 ≈ 16.97 cm².

例：三角形两边为 6 cm 和 8 cm，夹角为 45°。面积 = ½ × 6 × 8 × sin 45° = 24 × √2/2 ≈ 16.97 cm²。

This formula is often the fastest way to solve area problems in non-right triangles.

这个公式常是解决非直角三角形面积问题最快的方法。

7. Bearings | 方位角

A bearing is the angle measured clockwise from north. It is always written as three digits, e.g. 045°, 120°, 270°.

方位角是从正北方向顺时针量出的角度，通常用三位数表示，如 045°、120°、270°。

In trigonometry, bearings create right-angled or non-right triangles. You can use sine and cosine rules to find distances and directions.

在三角学中，方位角可构成直角三角形或非直角三角形。你可以用正弦、余弦定理求出距离和方向。

Example: A boat travels 50 km on a bearing of 060°, then 30 km on a bearing of 140°. The angle between the two paths is 80°, so you can calculate the final displacement using the cosine rule.

例：一艘船沿方位角 060° 行驶 50 km，再沿方位角 140° 行驶 30 km。两条路径夹角为 80°，可用余弦定理计算最终位移。

Always draw a clear diagram when dealing with bearings. Mark the north lines and the given angles carefully.

处理方位角时一定要画清楚示意图，标出北向线和已知角度。

8. Graphs of Trigonometric Functions | 三角函数图像

The graphs of sin θ, cos θ, and tan θ show periodic behavior. In IGCSE, you should recognise their shapes and key features.

sin θ、cos θ 和 tan θ 的图像呈现周期性。IGCSE 要求你识别它们的形状和关键特征。

y = sin θ: starts at (0, 0), maximum 1 at 90°, minimum −1 at 270°, period 360°.

y = sin θ：从 (0, 0) 出发，在 90° 取最大值 1，在 270° 取最小值 −1，周期为 360°。
y = cos θ: starts at (0, 1), mirrors a sine graph shifted 90° left.

y = cos θ：从 (0, 1) 出发，相当于正弦图像向左平移 90°。
y = tan θ: repeats every 180° and has vertical asymptotes at 90°, 270°, etc.

y = tan θ：每 180° 重复一次，并在 90°、270° 等处有垂直渐近线。

You should be able to sketch transformations such as y = a sin bx, where a changes amplitude and b changes period.

你还应会画简单变换图，如 y = a sin bx，其中 a 改变振幅，b 改变周期。

9. Solving Basic Trigonometric Equations | 解基本三角方程

In IGCSE Extended papers, you may be asked to solve equations like sin θ = 0.5 for 0° ≤ θ ≤ 360°.

在 IGCSE 拓展卷中，你可能需要解如 sin θ = 0.5 在 0° ≤ θ ≤ 360° 范围内的方程。

Step 1: find the acute reference angle using inverse sine: θ = sin⁻¹(0.5) = 30°.

第一步：用反正弦求锐角参考角：θ = sin⁻¹(0.5) = 30°。

Step 2: use the CAST diagram or graph to find all solutions in the required range. For sin positive, θ is in quadrant I and II, so θ = 30° and θ = 150°.

第二步：利用 CAST 图或图像找出范围内的所有解。sin 为正时，θ 位于第一、第二象限，因此 θ = 30° 和 θ = 150°。

For cos positive, quadrant I and IV; for tan positive, quadrant I and III. Remember the period: sin and cos repeat every 360°, tan every 180°.

cos 为正时，θ 位于第一、第四象限；tan 为正时，θ 位于第一、第三象限。注意周期：sin 和 cos 每 360° 重复，tan 每 180° 重复。

10. Common Mistakes to Avoid | 常见错误

Many students lose marks due to avoidable errors. Watch out for these:

许多学生因可避免的错误而失分。请特别留意以下几点：

Using degree mode when the question is in radians (or vice versa). IGCSE usually works in degrees, but always check.

题目用弧度但计算器用角度（或相反）。IGCSE 通常使用角度，但请务必检查。
Applying the sine rule or cosine rule to right-angled triangles when simple SOH CAH TOA is faster.

对直角三角形使用正弦/余弦定理，其实用 SOH CAH TOA 更快。
Forgetting to round correctly. IGCSE often asks for 1 decimal place or 3 significant figures.

忘记按要求取近似值。IGCSE 常要求保留 1 位小数或 3 位有效数字。
Mixing up opposite and adjacent sides in a non-right triangle diagram.

在非直角三角形中混淆对边和邻边。

Always draw the triangle, label the sides, and choose the correct formula before calculating.

计算前务必画出三角形、标注边长，并选择正确的公式。

11. Practice Questions | 练习

Try these questions to test your understanding:

尝试以下问题来检验你的理解：

In a right triangle, the hypotenuse is 13 cm and one angle is 23°. Find the length of the side opposite this angle.

在直角三角形中，斜边为 13 cm，一个角为 23°。求该角的对边长度。
Triangle ABC has AB = 7 cm, AC = 9 cm, and angle A = 48°. Find BC using the cosine rule.

三角形 ABC 中，AB = 7 cm，AC = 9 cm，角 A = 48°。用余弦定理求 BC。
Solve for θ when sin θ = −0.5, for 0° ≤ θ ≤ 360°.

解方程 sin θ = −0.5，其中 0° ≤ θ ≤ 360°。
A ship travels 20 km due east, then 15 km on a bearing of 135°. Find the distance from the starting point.

一艘船先向正东行驶 20 km，再沿方位角 135° 行驶 15 km。求它离起点的距离。

Answers: 1. 5.08 cm (approx) 2. 6.77 cm (approx) 3. 210°, 330° 4. 24.7 km (approx).

答案：1. 约 5.08 cm 2. 约 6.77 cm 3. 210°、330° 4. 约 24.7 km。

12. Summary and Exam Tips | 总结与考试提示

Trigonometry rewards practice. Know your formulas and when to use them:

三角学熟能生巧。记牢公式并知道何时使用：

Right triangle: SOH CAH TOA.

直角三角形：SOH CAH TOA。
Any triangle (two sides + included angle): Area = ½ ab sin C.

任意三角形（两边 + 夹角）：面积 = ½ ab sin C。
Any triangle (two angles + side, or two sides + non-included angle): Sine rule.

任意三角形（两角 + 一边，或两边 + 其中一边对角）：正弦定理。
Any triangle (two sides + included angle, or three sides): Cosine rule.

任意三角形（两边 + 夹角，或三边）：余弦定理。

Read the question carefully, identify which information is given, and choose the most direct route. Show clear working to earn method marks.

仔细读题，识别已知条件，选择最直接的解法。写出清晰步骤可获取方法分。

Finally, always check your calculator mode and whether your final answer is sensible for the diagram.

最后，随时检查计算器模式，并确认最终答案是否符合图形逻辑。

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Trigonometry for IGCSE: Ratios, Rules and Applications | IGCSE 三角函数：比例、定理与应用