Trigonometry Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：三角函数全考点精讲

📚 Trigonometry Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：三角函数全考点精讲

Trigonometry is a core topic in GCSE Edexcel Mathematics, linking angles and side lengths in right-angled and non-right-angled triangles. You will need to master the sine, cosine and tangent ratios, exact values for key angles, problem-solving with bearings, elevation and depression, and the sine rule, cosine rule and area formulas for any triangle. This comprehensive revision guide covers every essential concept, with clear explanations, worked examples and exam tips to help you secure top marks.

三角函数是 GCSE Edexcel 数学的核心考点，它将直角三角形和非直角三角形中的角度与边长联系起来。你需要掌握正弦、余弦和正切比，关键角度的精确值，方位角、仰角和俯角问题，以及适用于任意三角形的正弦定理、余弦定理和面积公式。这份全面的复习指南涵盖所有基本概念，提供清晰的解释、解题示例和考试技巧，帮助你稳夺高分。

1. Basic Trigonometric Ratios: SOH CAH TOA | 基础三角比：SOH CAH TOA

The three trigonometric ratios are defined on a right-angled triangle. For an angle θ, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent. The mnemonic SOH CAH TOA helps you recall these relationships quickly. Always label the sides relative to the angle you are working with: the hypotenuse is the longest side, the opposite is the side facing the angle, and the adjacent is the side next to the angle (not the hypotenuse).

三个三角比定义在直角三角形上。对于角 θ，sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记忆口诀 SOH CAH TOA 帮助你快速回想这些关系。始终根据你所处理的角来标注各边：斜边是最长边，对边是正对着角的边，邻边是与角相邻的边（不是斜边）。

sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent

sin θ = 对边 / 斜边
cos θ = 邻边 / 斜边
tan θ = 对边 / 邻边

Tip: When using your calculator, ensure it is in degree mode (D or DEG) for GCSE questions.

提示：使用计算器时，务必确保其处于角度模式（D 或 DEG），因为 GCSE 题目都用度数。

2. Exact Trigonometric Values for Key Angles | 特殊角的精确三角比值

You are expected to memorise the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90°. These often appear in non-calculator papers. The table below summarises them:

你需要熟记 0°、30°、45°、60° 和 90° 的正弦、余弦和正切的精确值。这些经常出现在非计算器试卷中。下表总结如下：

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

Notice the pattern: sin values increase from 0 to 1, cos values decrease from 1 to 0, and tan values rise from 0 to undefined. Drawing the two special triangles – the 45° right-angled isosceles (sides 1, 1, √2) and the 30°-60° right triangle (sides 1, √3, 2) – will help you derive these quickly if you forget them under pressure.

注意变化规律：sin 值从 0 增加到 1，cos 值从 1 减小到 0，tan 值从 0 增加到无穷（不存在）。画出两个特殊三角形——45° 的等腰直角三角形（边长 1, 1, √2）和 30°-60° 直角三角形（边长 1, √3, 2）——能帮助你在紧张时快速推导出这些值。

3. Finding a Missing Side Using Trigonometry | 利用三角函数求未知边长

When you know one acute angle and one side of a right-angled triangle, you can find an unknown side by choosing the appropriate ratio. For example, to find the adjacent side when you know the hypotenuse and angle, use cos θ = adjacent / hypotenuse, then rearrange to adjacent = hypotenuse × cos θ.

当你知道直角三角形的一个锐角和一条边时，你可以选择合适的三角比求出未知边。例如，若已知斜边和角求邻边，可用 cos θ = 邻边 / 斜边，再变形得到邻边 = 斜边 × cos θ。

Worked example: A ladder 5 m long leans against a wall, making an angle of 70° with the ground. Find how far the foot of the ladder is from the wall. The distance from the wall is the adjacent side to the 70° angle, the ladder is the hypotenuse. So adjacent = 5 × cos 70° ≈ 1.71 m (using a calculator).

解题示例：一架长 5 m 的梯子斜靠在墙上，与地面成 70° 角。求梯脚离墙的距离。梯脚到墙的距离是 70° 角的邻边，梯子是斜边。因此邻边 = 5 × cos 70° ≈ 1.71 m（使用计算器）。

Always label your triangle with O, A, H, choose the ratio that involves the known side and the side you want, then solve. Do not forget units.

始终用 O、A、H 标注三角形，选择包含已知边和所求边的三角比，然后求解。不要忘记单位。

4. Finding an Angle Using Inverse Trigonometry | 利用反三角函数求角度

If you know two sides of a right-angled triangle, you can find an acute angle by using the inverse functions sin⁻¹, cos⁻¹, tan⁻¹ (also written arcsin, arccos, arctan). For instance, if opposite = 3 and hypotenuse = 5, then sin θ = 3/5 → θ = sin⁻¹(3/5) ≈ 36.9°.

若已知直角三角形的两条边，你可以使用反函数 sin⁻¹、cos⁻¹、tan⁻¹（也写作 arcsin、arccos、arctan）求出锐角。例如，若对边 = 3，斜边 = 5，则 sin θ = 3/5 → θ = sin⁻¹(3/5) ≈ 36.9°。

These buttons are found above the sin, cos, tan keys on your calculator, usually accessed by pressing SHIFT or 2nd. Make sure you put the side ratio in brackets correctly. Typical exam questions will ask you to round your answer to one decimal place or to the nearest degree.

这些按钮在计算器的 sin、cos、tan 键上方，通常按 SHIFT 或 2nd 键进入。确保正确地将边长比值放入括号内。典型的试题会要求你将答案四舍五入到一位小数或最接近的度数。

Tip: Before pressing sin⁻¹, check that your ratio is between -1 and 1 (for sin and cos) or any real number (for tan). Otherwise you’ve misidentified sides.

提示：在按 sin⁻¹ 之前，检查比值是否在 -1 到 1 之间（sin 和 cos），tan 则不受限制。否则说明你认错了边。

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

Angles of elevation and depression are measured from the horizontal. The angle of elevation is the upward angle from the horizontal to a line of sight above. The angle of depression is the downward angle from the horizontal to a line of sight below. They appear in real-world problems involving cliffs, buildings, planes and boats.

仰角和俯角都是从水平线开始测量的。仰角是从水平线向上看物体的视线夹角。俯角是从水平线向下看物体的视线夹角。它们出现在涉及悬崖、建筑物、飞机和船只的实际问题中。

These angles are always formed with the horizontal line. A key fact: the angle of depression from a point A to a point B is equal to the angle of elevation from B to A, because they are alternate interior angles on parallel lines (horizontal lines at A and B). Draw a clear diagram, place the horizontal line, and identify the right-angled triangle involved.

这些角总是与水平线一起形成。一个关键事实：从点 A 到点 B 的俯角等于从点 B 到点 A 的仰角，因为它们是平行线（A 和 B 处的水平线）上的内错角。画出清晰示意图，放置水平线，并找出所涉及的直角三角形。

For example, from the top of a cliff 60 m high, a boat is seen at an angle of depression of 22°. The distance of the boat from the base of the cliff equals 60 / tan 22° (using tan = opposite / adjacent).

例如，从 60 m 高的悬崖顶部看到一艘船，俯角为 22°。船离悬崖底部的距离等于 60 / tan 22°（用 tan = 对边 / 邻边）。

6. Trigonometry in 3D | 三维空间中的三角函数

GCSE Edexcel also tests your ability to apply trigonometry in 3D shapes such as cuboids, prisms and pyramids. The trick is to identify or draw a relevant right-angled triangle within the solid. You often need to use Pythagoras’ theorem first to find a missing length (e.g., a diagonal inside a face or the body diagonal), and then use trigonometry to find an angle, for example the angle between a line and a plane.

GCSE Edexcel 也考察你在长方体、棱柱和棱锥等三维形状中应用三角函数的能力。关键在于在立体中找出或画出相关的直角三角形。你通常需要先用勾股定理求出未知长度（例如，面上的对角线或体对角线），然后再用三角函数求角，比如线与平面之间的夹角。

Worked example: In a cuboid with dimensions 3 cm × 4 cm × 5 cm, find the angle between the body diagonal and the base. First calculate the base diagonal: √(3² + 4²) = 5 cm. Then the triangle formed by the body diagonal, the base diagonal and the height (5 cm). tan θ = height / base diagonal = 5/5 = 1 → θ = 45°.

解题示例：在一个尺寸为 3 cm × 4 cm × 5 cm 的长方体中，求体对角线与底面之间的夹角。先计算底面对角线：√(3² + 4²) = 5 cm。然后由体对角线、底面对角线和高（5 cm）形成的三角形中，tan θ = 高 / 底面对角线 = 5/5 = 1 → θ = 45°。

Always draw the triangle flat on your paper, label the lengths, and then apply SOH CAH TOA. For the angle between a line and a plane, project the line onto the plane and take the angle between the line and its projection.

始终将三角形画在纸上，标注边长，然后应用 SOH CAH TOA。对于线与平面的夹角，将线投影到平面上，然后取该线与投影线之间的夹角。

7. The Sine Rule | 正弦定理

For any triangle (not necessarily right-angled), the sine rule states: a / sin A = b / sin B = c / sin C, where sides a, b, c are opposite angles A, B, C respectively. Alternatively, it can be written as sin A / a = sin B / b = sin C / c. Use this rule when you know: two angles and any side (AAS or ASA), or two sides and a non-included angle (SSA).

对于任意三角形（不一定是直角三角形），正弦定理表示为：a / sin A = b / sin B = c / sin C，其中边 a、b、c 分别对应角 A、B、C。也可以写成 sin A / a = sin B / b = sin C / c。当你已知：两角一边（AAS 或 ASA）或两边及一个非夹角（SSA）时，使用正弦定理。

Be careful with the ambiguous case (SSA) when finding an angle. For example, if you are given sides and a non-included angle, there may be two possible triangles (one acute angle and one obtuse angle with the same sine value). In GCSE, you typically decide on the correct triangle based on the context (angles in a triangle sum to 180°).

求角时要小心不明确情况（SSA）。例如，若已知两边和一个非夹角，可能有两种可能的三角形（一个锐角和一个钝角具有相同的正弦值）。在 GCSE 中，通常根据上下文（三角形内角和为 180°）判断正确的解。

Example: In triangle ABC, a = 8 cm, b = 10 cm, A = 30°. Find angle B. Using sin B / b = sin A / a → sin B = (10 × sin 30°)/8 = 0.625 → B ≈ 38.7° or 141.3°. Since the sum of angles would exceed 180° if B = 141.3° (with A = 30°), we take B = 38.7°.

示例：在三角形 ABC 中，a = 8 cm，b = 10 cm，A = 30°。求角 B。由 sin B / b = sin A / a → sin B = (10 × sin 30°)/8 = 0.625 → B ≈ 38.7° 或 141.3°。由于若 B = 141.3°（加 A = 30°）则角度和超过 180°，所以取 B = 38.7°。

8. The Cosine Rule | 余弦定理

The cosine rule links the three sides of a triangle with the cosine of one angle: a² = b² + c² – 2bc cos A. It can also be rearranged to find an angle: cos A = (b² + c² – a²) / (2bc). Use the cosine rule when you know: three sides (to find an angle), or two sides and the included angle (SAS) to find the third side.

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。也可以变形求角：cos A = (b² + c² – a²) / (2bc)。当你已知：三边（求角）或两边及其夹角（SAS）求第三边时，使用余弦定理。

Always label the side opposite the angle you are finding or using as ‘a’, and the other two sides as ‘b’ and ‘c’. When calculating an angle, you will need to use the inverse cosine function (cos⁻¹). The cosine rule is particularly useful in bearings and triangle navigation problems.

始终将你要求或使用的角所对的边标记为 ‘a’，其他两边标记为 ‘b’ 和 ‘c’。计算角度时，你需要使用反余弦函数（cos⁻¹）。余弦定理在方位角和三角形导航问题中特别有用。

Example: In triangle ABC, a = 7 cm, b = 5 cm, c = 8 cm. Find angle A. cos A = (5² + 8² – 7²) / (2 × 5 × 8) = (25+64-49)/80 = 40/80 = 0.5 → A = cos⁻¹(0.5) = 60°.

示例：在三角形 ABC 中，a = 7 cm，b = 5 cm，c = 8 cm。求角 A。cos A = (5² + 8² – 7²) / (2 × 5 × 8) = (25+64-49)/80 = 40/80 = 0.5 → A = cos⁻¹(0.5) = 60°。

9. Area of a Triangle Using ½ab sin C | 使用 ½ab sin C 求三角形面积

The area of any triangle can be found using the formula: Area = ½ ab sin C, where a and b are two sides and C is the angle between them. This formula eliminates the need to find the perpendicular height in many non-right-angled triangles.

任意三角形的面积可以用公式：面积 = ½ ab sin C 求出，其中 a 和 b 是两边，C 是这两边的夹角。这个公式省去了在许多非直角三角形中寻找垂直高度的需要。

Essentially, you multiply half the product of two sides by the sine of the included angle. If you know two sides and the angle between them, you can directly calculate the area. If you know three sides, use the cosine rule first to find an angle, then apply the area formula.

本质上，你将两边的乘积的一半乘以夹角的正弦。如果你知道两边及其夹角，可以直接计算面积。如果知道三边，先用余弦定理求出一个角，再应用面积公式。

Example: Triangle sides 9 cm and 12 cm with included angle 50°. Area = ½ × 9 × 12 × sin 50° ≈ 41.4 cm².

示例：三角形两边为 9 cm 和 12 cm，夹角 50°。面积 = ½ × 9 × 12 × sin 50° ≈ 41.4 cm²。

Remember to label angles in degrees and round appropriately. The formula also underpins the proof of the sine rule, but you just need to use it in problem-solving.

记住角度用度数，并适当四舍五入。该公式也是正弦定理证明的基础，但你在解题中只需直接应用。

10. Trigonometric Graphs and Simple Equations | 三角函数图像与简单方程

It’s important to recognise the shapes 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°, drops to -1 at 270° and returns to 0 at 360°, creating a wave. The cosine graph starts at 1 at 0°, falls to 0 at 90°, to -1 at 180°, back to 0 at 270° and to 1 at 360°. The tangent graph has asymptotes at 90° and 270°, repeating every 180°, with values from -∞ to +∞.

认识 y = sin x、y = cos x 和 y = tan x 在 0° ≤ x ≤ 360° 区间内的形状很重要。正弦图像从 0 开始，到 90° 升至 1，到 180° 降至 0，到 270° 降至 -1，到 360° 回到 0，形成波浪。余弦图像从 0° 的 1 开始，到 90° 降至 0，到 180° 降至 -1，到 270° 回到 0，到 360° 回到 1。正切图像在 90° 和 270° 有渐近线，每 180° 重复一次，取值从 -∞ 到 +∞。

You may be asked to solve equations like sin x = 0.5 for 0° ≤ x ≤ 360°. First find the principal value using your calculator: sin⁻¹(0.5) = 30°. Then use the symmetry of the sine graph to find the second solution: sin(180° – θ) = sin θ, so another answer is 180° – 30° = 150°. For cosine, cos(360° – θ) = cos θ. For tangent, tan(180° + θ) = tan θ.

你可能会被要求解方程，比如在 0° ≤ x ≤ 360° 范围内求 sin x = 0.5 的解。先用计算器求主值：sin⁻¹(0.5) = 30°。然后利用正弦图像的对称性找第二个解：sin(180° – θ) = sin θ，所以另一个解是 180° – 30° = 150°。对于余弦，cos(360° – θ) = cos θ。对于正切，tan(180° + θ) = tan θ。

Familiarity with the graphs helps you avoid missing solutions. Always check the range given in the question.

熟悉图像有助于避免漏解。一定要检查题目给定的区间。

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Trigonometry Revision for GCSE Edexcel Maths | GCSE Edexcel 数学：三角函数全考点精讲