GCSE WJEC Mathematics: Trigonometry – Key Points | GCSE WJEC 数学：三角函数考点精讲

📚 GCSE WJEC Mathematics: Trigonometry – Key Points | GCSE WJEC 数学：三角函数考点精讲

Trigonometry is a core topic in the GCSE WJEC Mathematics Higher tier, covering the relationships between the sides and angles of triangles. It is essential for solving problems involving right-angled triangles, non-right-angled triangles, bearings, and three-dimensional shapes. This article provides a detailed walkthrough of the key points, formulas, and common question types to help you prepare for the exam.

三角函数是 GCSE WJEC 数学高级卷的核心主题，研究三角形边与角之间的关系。它是解决直角三角形、非直角三角形、方位角和三维图形问题的关键。本文详细梳理了考点、公式和常见题型，帮助你备考。

1. Introduction to Trigonometric Ratios | 三角比介绍

In a right-angled triangle, the three basic trigonometric ratios relate an acute angle to the lengths of two sides. They are defined as sine (sin), cosine (cos), and tangent (tan). For a given angle θ, sinθ = opposite / hypotenuse, cosθ = adjacent / hypotenuse, tanθ = opposite / adjacent.

在直角三角形中，三个基本三角比将一个锐角与两条边的长度联系起来。它们定义为正弦（sin）、余弦（cos）和正切（tan）。对于给定角 θ，sinθ = 对边 / 斜边，cosθ = 邻边 / 斜边，tanθ = 对边 / 邻边。

The opposite side is the side directly facing the angle, the adjacent side is next to the angle and is not the hypotenuse, and the hypotenuse is the longest side, always opposite the right angle.

对边是正对着角的那条边，邻边是与角相邻且不是斜边的那条边，斜边是最长的边，始终对着直角。

A common mnemonic to remember these is ‘SOH CAH TOA’: Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent.

记住这些关系的常用口诀是“SOH CAH TOA”：Sin = 对边/斜边，Cos = 邻边/斜边，Tan = 对边/邻边。

2. Special Angles and Exact Values | 特殊角与精确值

WJEC expects you to know the exact trigonometric values for 0°, 30°, 45°, 60°, and 90° without using a calculator. These values are derived from isosceles right triangles (45°) and half-equilateral triangles (30° and 60°).

WJEC 要求你牢记 0°、30°、45°、60° 和 90° 的精确三角函数值，无需使用计算器。这些值来源于等腰直角三角形（45°）和半等边三角形（30° 和 60°）。

The exact values are as follows:

精确值如下：

θ	sinθ	cosθ	tanθ
0°	0	1	0
30°	½	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	½	√3
90°	1	0	undefined

Using these values, you can evaluate expressions like sin²30° + cos²30° = (½)² + (√3/2)² = ¼ + ¾ = 1.

利用这些值，你可以计算诸如 sin²30° + cos²30° = (½)² + (√3/2)² = ¼ + ¾ = 1 的表达式。

3. Solving Right-Angled Triangles | 解直角三角形

To find an unknown side in a right-angled triangle, identify the given angle and sides, choose the appropriate ratio (SOH CAH TOA), set up an equation, and solve. For example, to find the opposite side when you know the adjacent side and the angle, use tanθ = opposite / adjacent.

要在直角三角形中求未知边，先确定已知角和边，选择合适的三角比（SOH CAH TOA），列出方程并求解。例如，若已知邻边和角求对边，用 tanθ = 对边 / 邻边。

To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) on your calculator. For instance, if sinθ = 0.5, then θ = sin⁻¹(0.5) = 30°.

要求未知角，使用计算器上的反三角函数（sin⁻¹, cos⁻¹, tan⁻¹）。例如，若 sinθ = 0.5，则 θ = sin⁻¹(0.5) = 30°。

Always check that your calculator is in degree mode when working with angles in degrees.

使用角度制时，务必确保计算器处于度数模式。

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

The angle of elevation is the angle measured upwards from a horizontal line to the line of sight. The angle of depression is measured downwards from a horizontal line. These two angles are equal when lines are parallel.

仰角是从水平线向上测量到视线的角。俯角是从水平线向下测量的角。当两条线平行时，仰角与俯角相等。

Typical WJEC problems involve a person looking at the top of a building (elevation) or from a cliff looking down at a boat (depression). You draw a right-angled triangle and apply trigonometric ratios.

WJEC 的典型问题包括人仰望建筑顶部（仰角）或从悬崖俯视船只（俯角）。画出直角三角形，然后应用三角比。

Label the horizontal distance, vertical height, and line of sight as the adjacent, opposite, and hypotenuse respectively, then use SOH CAH TOA.

将水平距离、垂直高度和视线分别标记为邻边、对边和斜边，再使用 SOH CAH TOA。

5. Bearings | 方位角

Bearings are used in navigation to describe direction. A bearing is measured clockwise from North, always given as a three-figure number (e.g., 035°, 270°).

方位角用于导航中描述方向。方位角从正北起顺时针测量，始终以三位数表示（如 035°、270°）。

Trigonometry problems with bearings often ask you to find the distance between two points or the bearing of one point from another. Construct a right-angled triangle using North-South and East-West lines.

涉及方位角的三角学问题常要求计算两点之间的距离或一点相对于另一点的方位角。利用南北线和东西线构造直角三角形。

For example, a ship sails on a bearing of 060° for 10 km. Its Northward distance is 10 × cos60° = 5 km, and its Eastward distance is 10 × sin60° = 10 × √3/2 ≈ 8.66 km.

例如，一艘船以 060° 的方位角航行 10 公里。它向北的距离为 10 × cos60° = 5 公里，向东的距离为 10 × sin60° = 10 × √3/2 ≈ 8.66 公里。

6. The Sine Rule | 正弦定理

The Sine Rule applies to any triangle, not just right-angled ones. It states that a / sinA = b / sinB = c / sinC, where a, b, c are side lengths opposite angles A, B, C respectively.

正弦定理适用于任何三角形，不限于直角三角形。它表述为 a / sinA = b / sinB = c / sinC，其中 a、b、c 分别是对角 A、B、C 的边长。

Use the Sine Rule when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA). Be cautious with the SSA case, as it may produce an ambiguous solution (two possible triangles).

当已知两角一边（AAS 或 ASA）或两边及一个非夹角（SSA）时，可使用正弦定理。注意 SSA 情况可能产生歧义（两个可能的三角形），需小心处理。

To find an angle using the Sine Rule, write sinA / a = sinB / b and solve for the unknown angle. Always check if the supplementary angle (180° − θ) is also valid in the context of the triangle’s angle sum.

要用正弦定理求角，列出 sinA / a = sinB / b 并解出未知角。始终检查补角（180° − θ）在三角形内角和中是否也有效。

7. The Cosine Rule | 余弦定理

The Cosine Rule links the three sides and one angle of any triangle. It has two useful forms: a² = b² + c² − 2bc cosA to find a side, and cosA = (b² + c² − a²) / (2bc) to find an angle.

余弦定理将任意三角形的三边和一个角联系起来。它有两种常用形式：a² = b² + c² − 2bc cosA 用于求边，cosA = (b² + c² − a²) / (2bc) 用于求角。

Apply the Cosine Rule when you have two sides and the included angle (SAS) and need the third side, or when you know all three sides (SSS) and need an angle.

当已知两边及其夹角（SAS）求第三边，或已知三边（SSS）求角时，应用余弦定理。

For example, in a triangle with sides b = 7, c = 8, and angle A = 60°, find side a: a² = 7² + 8² − 2×7×8×cos60° = 49 + 64 − 112×0.5 = 113 − 56 = 57, so a = √57 ≈ 7.55.

例如，在三角形中 b=7, c=8, 角 A=60°，求边 a：a² = 7² + 8² − 2×7×8×cos60° = 49 + 64 − 112×0.5 = 113 − 56 = 57，因此 a = √57 ≈ 7.55。

8. Area of a Triangle | 三角形面积

Besides the standard ½ × base × height formula, the area of any triangle can be found using two sides and the included angle: Area = ½ ab sinC (or ½ bc sinA, ½ ac sinB).

除了标准的 ½ × 底 × 高公式，任意三角形的面积还可用两边及其夹角计算：面积 = ½ ab sinC（或 ½ bc sinA、½ ac sinB）。

This formula is particularly useful when the perpendicular height is not given directly. It frequently appears in WJEC questions alongside the Sine and Cosine Rules, especially in compound shapes.

该公式在未直接给出垂直高度时特别有用。它常与正弦定理和余弦定理一起出现在 WJEC 考试题中，尤其是在复合图形问题中。

Always use the included angle between the two known sides. If you are given an angle that is not included, find the necessary side or angle first using the Sine or Cosine Rule.

一定要使用两条已知边之间的夹角。若给出的角不是夹角，先用正弦定理或余弦定理求出所需的边或角。

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

You need to recognise and sketch the graphs of y = sin x, y = cos x, and y = tan x for 0° ≤ x ≤ 360°. The sine graph starts at 0, rises to 1 at 90°, falls to 0 at 180°, goes to −1 at 270°, and returns to 0 at 360°. It is periodic with a period of 360°.

你需要识别并画出 y = sin x、y = cos x 和 y = tan x 在 0° ≤ x ≤ 360° 的图像。正弦图像从 0 开始，在 90° 升至 1，在 180° 回到 0，在 270° 降至 −1，在 360° 回到 0。其周期为 360°。

The cosine graph starts at 1, falls to 0 at 90°, goes to −1 at 180°, returns to 0 at 270°, and ends at 1 at 360°. The tangent graph has asymptotes at 90° and 270°, where it is undefined, and its period is 180°.

余弦图像从 1 开始，在 90° 降至 0，在 180° 到 −1，在 270° 回到 0，在 360° 结束于 1。正切图像在 90° 和 270° 有渐近线，此处无定义，其周期为 180°。

WJEC may ask you to use these graphs to solve simple trigonometric equations, such as sin x = 0.5 for x between 0° and 360°. You would read off the solutions: x = 30° and x = 150°, using symmetry.

WJEC 可能会要求你利用这些图像解简单的三角方程，例如在 0° 到 360° 之间解 sin x = 0.5。通过对称性读出解：x = 30° 和 x = 150°。

10. 3D Trigonometry | 三维三角函数

3D trigonometry problems involve finding lengths or angles in three-dimensional shapes such as cuboids, pyramids, and prisms. You must identify right-angled triangles within the 3D figure, often using Pythagoras’ theorem first to find a diagonal or slant height.

三维三角函数问题涉及在长方体、棱锥和棱柱等三维图形中求长度或角度。你必须在三维图形中识别出直角三角形，通常需先用勾股定理求出对角线或斜高。

A common WJEC question gives you a cuboid and asks for the angle between the diagonal of the base and the plane of the base, or the angle between a face diagonal and the vertical. Break the problem into two steps: find the required lengths, then apply SOH CAH TOA.

WJEC 常见题型是给出一个长方体，求底面对角线与底面平面之间的角，或面对角线与垂线之间的角。将问题分解为两步：求出所需长度，然后应用 SOH CAH TOA。

For a square-based pyramid, you might need the angle between a slant edge and the base. This requires the perpendicular height and the horizontal distance from the centre to a vertex, which can be found using half the diagonal of the base.

对于正四棱锥，你可能需要求侧棱与底面之间的角。这需要垂直高度和从中心到顶点的水平距离，该距离可通过底面对角线的一半求得。

11. Trigonometric Identities and Equations | 三角恒等式与方程

At the WJEC Higher tier, you are expected to know and use the identity sin²θ + cos²θ = 1. This identity holds true for all values of θ and can be used to find one trigonometric ratio when another is given, or to prove other equations.

在 WJEC 高级卷中，你需要知道并使用恒等式 sin²θ + cos²θ = 1。该恒等式对所有 θ 值都成立，可用于在已知一个三角比时求另一个，或证明其他等式。

You may also be asked to solve linear trigonometric equations such as 2sin x = 1 for x between 0° and 360°. Use the graph or quadrant rule to find all solutions: first find the principal angle, then apply the symmetry of the sine, cosine, or tangent.

你也可能被要求解线性三角方程，例如在 0° 到 360° 内解 2sin x = 1。利用图像或象限法则找出所有解：先求出主角，再应用正弦、余弦或正切的对称性。

When solving equations involving cosine, remember that cos x = cos(360° − x). For sine, sin x = sin(180° − x). For tangent, the period is 180°, so tan x = tan(x + 180°).

解含余弦的方程时，记住 cos x = cos(360° − x)。对于正弦，sin x = sin(180° − x)。对于正切，周期为 180°，因此 tan x = tan(x + 180°)。

Practise rearranging equations to isolate the trigonometric function before solving. For example, 3cos²x – cos x – 2 = 0 can be treated as a quadratic in cos x, factorised, and solved for the allowable range.

练习在求解前将方程变形以分离三角函数。例如，3cos²x – cos x – 2 = 0 可视为关于 cos x 的二次方程，通过因式分解后在允许范围内求解。

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GCSE WJEC Mathematics: Trigonometry – Key Points | GCSE WJEC 数学：三角函数 考点精讲