IGCSE WJEC Mathematics: Trigonometry | IGCSE WJEC 数学：三角函数考点精讲

📚 IGCSE WJEC Mathematics: Trigonometry | IGCSE WJEC 数学：三角函数考点精讲

Trigonometry is a key topic in IGCSE WJEC Mathematics, involving the study of relationships between angles and side lengths in triangles. This revision guide covers trigonometric ratios in right-angled triangles, exact values for special angles, solving for missing sides and angles, angles of elevation and depression, the sine and cosine rules, the area formula using sine, and graphs of trigonometric functions. Each concept is explained with clear examples and tips to help you succeed in your exam.

三角函数是 IGCSE WJEC 数学中的一个重要考点，研究三角形中角度与边长之间的关系。本篇复习指南涵盖直角三角形中的三角函数比、特殊角的精确值、求解未知边与未知角、仰角与俯角、正弦定理与余弦定理、利用正弦求三角形面积的公式以及三角函数图像。每个概念都配有清晰的解析和提示，帮助你在考试中取得好成绩。

1. Trigonometric Ratios in Right-Angled Triangles | 直角三角形中的三角函数比

In any right-angled triangle, the sides are named relative to a chosen acute angle: the hypotenuse is the longest side opposite the right angle, the opposite side is across from the given angle, and the adjacent side is next to the angle but not the hypotenuse.

在任意直角三角形中，根据所选锐角命名各边：斜边是直角对面最长的边，对边是给定角对面的边，邻边是与该角相邻但不是斜边的边。

The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). For an acute angle θ, they are defined as: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.

三个主要的三角函数比分别是正弦（sin）、余弦（cos）和正切（tan）。对于锐角 θ，它们的定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

These ratios remain constant for a given angle regardless of the size of the triangle, as long as the triangles are similar.

只要三角形相似，这些比值对于给定的角度保持不变，与三角形的大小无关。

2. SOH CAH TOA – Memory Aid | SOH CAH TOA 口诀

The acronym SOH CAH TOA is a simple way to memorise the trigonometric ratios: SOH stands for Sin = Opposite / Hypotenuse, CAH for Cos = Adjacent / Hypotenuse, and TOA for Tan = Opposite / Adjacent.

缩写 SOH CAH TOA 是记忆三角函数比的简单方法：SOH 代表 Sin = 对边/斜边，CAH 代表 Cos = 邻边/斜边，TOA 代表 Tan = 对边/邻边。

When labelling a right-angled triangle, first identify the right angle, then pick the acute angle you are interested in. Label the sides correctly before applying SOH CAH TOA.

在标记直角三角形时，先确定直角，然后选出你要研究的锐角。在应用 SOH CAH TOA 之前，正确标注三角形的各边。

3. Exact Values for Special Angles | 特殊角的精确值

WJEC IGCSE requires you to know the exact trigonometric values for 0°, 30°, 45°, 60°, and 90° without a calculator. These can be derived from two special triangles: the isosceles right-angled triangle (45°-45°-90°) and the equilateral triangle split in half (30°-60°-90°).

WJEC IGCSE 要求你无需使用计算器就能掌握 0°、30°、45°、60° 和 90° 这些特殊角的精确三角函数值。这些值可以从两个特殊三角形推出：等腰直角三角形（45°-45°-90°）和等边三角形的一半（30°-60°-90°）。

Angle θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Note that for 90°, tan θ is undefined because it would involve division by zero. These exact values frequently appear in non-calculator exam questions.

注意，对于 90°，tan θ 未定义，因为这涉及除以零。在非计算器考题中，这些精确值经常出现。

4. Finding Missing Sides | 求未知边长

To find a missing side in a right-angled triangle, choose the appropriate ratio based on the known angle and the given side. Write an equation and solve for the unknown.

要在直角三角形中求未知边长，根据已知角度和已知边选择合适的比，写出方程并求解未知数。

For example, given an angle of 40° and the adjacent side of 8 cm, to find the hypotenuse use cos: cos 40° = 8 / hypotenuse, so hypotenuse = 8 / cos 40°.

例如，已知一个角为 40°，邻边为 8 cm，要求斜边则使用余弦：cos 40° = 8 / 斜边，因此斜边 = 8 / cos 40°。

Always check that your calculator is in degree mode when solving such problems.

解此类题目时，务必检查计算器是否处于角度模式。

5. Finding Missing Angles | 求未知角度

When you need to find an acute angle in a right-angled triangle, use the inverse trigonometric functions: sin⁻¹, cos⁻¹, tan⁻¹ (also written as arcsin, arccos, arctan).

当需要求直角三角形中的锐角时，使用反三角函数：sin⁻¹, cos⁻¹, tan⁻¹（也写作 arcsin, arccos, arctan）。

For instance, if the opposite side is 5 cm and the hypotenuse is 13 cm, then sin θ = 5/13, so θ = sin⁻¹(5/13). Always round your answer to a given degree of accuracy, usually one decimal place.

例如，若对边为 5 cm，斜边为 13 cm，则 sin θ = 5/13，所以 θ = sin⁻¹(5/13)。答案通常要按要求精确到指定的小数位数，一般保留一位小数。

Questions may also test the ability to find an angle in a composite shape or a real-life context; break down the problem into right-angled triangles first.

题目也可能考查在复合图形或实际情景中求角度；先将问题分解为直角三角形再进行求解。

6. Angles of Elevation and Depression | 仰角与俯角

The angle of elevation is the angle measured upwards from the horizontal line to the line of sight when looking at an object above. The angle of depression is measured downwards from the horizontal to the line of sight when looking at an object below.

仰角是从水平线向上看向上方物体时，视线与水平线之间的夹角。俯角是从水平线向下看向下方物体时，视线与水平线之间的夹角。

These angles are always measured from the horizontal and are equal when the line of sight between two objects is parallel to the horizontal. In an examination, drawing a clear diagram with the horizontal line and labelling the angle will help you avoid confusion.

这些角总是从水平线开始测量，并且当两个物体之间的视线平行于水平线时，仰角和俯角相等。在考试中，画出清晰的示意图、标出水平线和角度有助于避免混淆。

Apply SOH CAH TOA using the identified right angle formed by the object’s vertical height, the horizontal distance, and the line of sight.

利用由物体垂直高度、水平距离和视线构成的直角三角形，应用 SOH CAH TOA 进行求解。

7. The Sine Rule | 正弦定理

The sine rule applies to any triangle, not just right-angled ones. It states that the ratio of a side length to the sine of its opposite angle is constant: a / sin A = b / sin B = c / sin C.

正弦定理适用于任意三角形，而不仅仅是直角三角形。它表明边长与其对角的正弦值之比是常数：a / sin A = b / sin B = c / sin C。

Use the sine rule when you know either two angles and one side (to find a missing side) or two sides and a non-included angle (to find a missing angle). Be aware of the ambiguous case when using the inverse sine: the angle could be acute or obtuse, but for WJEC IGCSE the questions usually make it clear which you need.

在已知两角一边（求未知边）或两边及其不夹角（求未知角）的情况下，使用正弦定理。使用反正弦函数时要注意模糊情况：角度可能为锐角或钝角，但在 WJEC IGCSE 考试中，题目通常会明确要求哪类答案。

For example, in triangle ABC, if A = 50°, B = 70° and a = 8 cm, you can find b using b = a sin B / sin A.

例如，在三角形 ABC 中，若 A = 50°，B = 70°，a = 8 cm，则可用 b = a sin B / sin A 求出 b。

8. The Cosine Rule | 余弦定理

The cosine rule links the three sides of a triangle to the cosine of one of its angles: a² = b² + c² − 2bc cos A. It can also be rearranged to find an angle: cos A = (b² + c² − a²) / 2bc.

余弦定理将三角形的三条边与其中一个角的余弦联系起来：a² = b² + c² − 2bc cos A。该公式也可变形为求角度：cos A = (b² + c² − a²) / 2bc。

Use the cosine rule when you have two sides and the included angle (to find the third side) or when you know all three sides (to find any angle).

当你已知两边及其夹角（求第三边）或已知三边求任意角时，使用余弦定理。

If a triangle has sides of 7 cm, 9 cm, and an included angle of 65°, the third side can be found by substituting into the formula.

若一个三角形的两边为 7 cm 和 9 cm，其夹角为 65°，则可通过代入公式求出第三边。

Remember that the side a is always opposite the angle A, so label your triangle consistently when applying this rule.

记住，边 a 总是对角 A 的对边，因此应用这一定理时必须一致标注三角形。

9. Area of a Triangle (1/2 ab sin C) | 三角形的面积公式 (1/2 ab sin C)

The area of any triangle can be computed using the formula Area = ½ × a × b × sin C, where a and b are two sides and C is the included angle between them.

任意三角形的面积可以利用公式面积 = ½ × a × b × sin C 进行计算，其中 a 和 b 是两条边，C 是它们之间的夹角。

This is particularly useful when the perpendicular height is not known. For example, if two sides of 5 cm and 8 cm form an angle of 40°, the area is ½ × 5 × 8 × sin 40°.

当不知道垂直高度时，这个公式特别有用。例如，若两边分别为 5 cm 和 8 cm，夹角为 40°，则面积为 ½ × 5 × 8 × sin 40°。

Make sure you are using the angle that lies directly between the two chosen sides, not the opposite angle.

要确保使用的是所选两条边之间的夹角，而不是对角。

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

The graphs of y = sin x, y = cos x and y = tan x are important for higher-tier candidates. The sine and cosine graphs are periodic with a period of 360°, while the tangent graph has a period of 180° and features vertical asymptotes every 90° from its starting point.

对于高阶考生而言，y = sin x，y = cos x 和 y = tan x 的图像至关重要。正弦和余弦图像是周期为 360° 的周期函数，而正切图像的周期为 180°，且从起点开始每 90° 有一条垂直渐近线。

y = sin x: starts at 0, reaches a maximum of 1 at 90°, returns to 0 at 180°, reaches a minimum of -1 at 270°, and back to 0 at 360°.

y = sin x：从 0 开始，在 90° 处达到最大值 1，在 180° 回到 0，在 270° 达到最小值 -1，在 360° 回到 0。

y = cos x: starts at 1, drops to 0 at 90°, reaches -1 at 180°, returns to 0 at 270°, and back to 1 at 360°.

y = cos x：从 1 开始，在 90° 处降至 0，在 180° 达到 -1，在 270° 回到 0，在 360° 回到 1。

y = tan x: crosses the origin, has vertical asymptotes at 90°, 270°, etc., and repeats every 180°.

y = tan x：经过原点，在 90°、270° 等处有垂直渐近线，每 180° 重复一次。

Understanding these shapes helps in solving trigonometric equations and interpreting transformations.

理解这些图像形状有助于求解三角函数方程和解读图像变换。

11. Applications and Exam Tips | 应用与应试技巧

Real-world problems often combine trigonometry with bearings, three-dimensional figures, or area calculations. Always draw a clear diagram, label known values, and identify which trig rule is best suited.

实际应用题常将三角函数与方位角、三维图形或面积计算相结合。务必画出清晰示意图，标出已知数值，并确定最合适的三角公式。

Check your answer’s reasonableness: side lengths must be positive, and angles in triangles must sum to 180°. If a calculation produces an impossible result, you may have used the wrong ratio or forgotten to convert units.

检查答案的合理性：边长必须为正，三角形内角和必须等于 180°。如果计算结果无法成立，可能使用了错误的比值或忘记换算单位。

Use exact values in non-calculator questions to simplify expressions, and never mix up sine and cosine rules: the sine rule involves opposite pairs, while the cosine rule connects three sides with one angle.

在非计算器题目中使用精确值以简化表达式，切勿混淆正弦定理和余弦定理：正弦定理涉及对边与对角，而余弦定理将三条边与一个角联系起来。

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IGCSE WJEC Mathematics: Trigonometry | IGCSE WJEC 数学：三角函数 考点精讲