GCSE OCR Maths: Trigonometry Key Points Explained | GCSE OCR 数学：三角函数 考点精讲

📚 GCSE OCR Maths: Trigonometry Key Points Explained | GCSE OCR 数学：三角函数 考点精讲

Trigonometry is a fundamental topic in the GCSE OCR Mathematics syllabus. It involves the study of relationships between side lengths and angles in triangles, with wide applications in solving real-world problems such as navigation, architecture, and engineering. This revision guide covers key concepts, essential formulas, and exam techniques to help you master trigonometric problems.

三角函数是 GCSE OCR 数学大纲中的一个基础课题。它研究三角形边长与角度之间的关系，广泛应用于导航、建筑和工程等实际问题的求解。本复习指南涵盖核心概念、必备公式以及应试技巧，助你攻克三角难题。

1. Right-Angled Triangle Trigonometry (SOH CAH TOA) | 直角三角形三角函数 (SOH CAH TOA)

In a right-angled triangle, the three trigonometric ratios relate an acute angle θ to the sides. The mnemonic SOH CAH TOA helps you remember which ratio uses which sides.

在直角三角形中，三个三角比将一个锐角 θ 与三边联系起来。记忆口诀 SOH CAH TOA 能帮你记住每个比例使用哪两条边。

sin θ = Opposite / Hypotenuse

正弦 = 对边 / 斜边

cos θ = Adjacent / Hypotenuse

余弦 = 邻边 / 斜边

tan θ = Opposite / Adjacent

正切 = 对边 / 邻边

The hypotenuse is always the longest side, opposite the right angle. The opposite side is opposite the given angle θ, and the adjacent side is next to θ (but not the hypotenuse).

斜边总是最长的边，对着直角。对边是给定角 θ 所对的边，邻边是与 θ 相邻的边（不是斜边）。

2. Finding Sides Using Trigonometry | 用三角函数求边长

To find an unknown side in a right-angled triangle, first label the sides relative to the given angle. Identify which two sides are involved (opposite, adjacent, hypotenuse) and choose the appropriate ratio (sin, cos, or tan). Set up the equation and solve for the missing side.

要求直角三角形的未知边长，首先根据给定角标注三边。确定涉及哪两条边（对边、邻边、斜边），选择合适的三角比（正弦、余弦或正切）。列出方程并求解未知边。

Example: If angle = 30°, hypotenuse = 10 cm, find the opposite side. Use sin 30° = opposite / 10 → opposite = 10 × sin 30°.

示例：若角度为 30°，斜边为 10 cm，求对边。使用 sin 30° = 对边/10 → 对边 = 10 × sin 30°。

Always check that your calculator is in degree mode. Look for a ‘DEG’ symbol or switch from radians if needed.

务必确保计算器处于角度（degree）模式。寻找 ‘DEG’ 符号，或根据需要从弧度模式切换。

3. Finding Angles Using Trigonometry | 用三角函数求角度

To find an unknown angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). Identify the two known sides, form the ratio, and apply the inverse function. For example, if opposite = 5 and hypotenuse = 8, then sin θ = 5/8, so θ = sin⁻¹(5/8).

求未知角度时，使用反三角函数（sin⁻¹、cos⁻¹、tan⁻¹）。确定已知的两条边，构成比值，然后应用反函数。例如，若对边为 5，斜边为 8，则 sin θ = 5/8，从而 θ = sin⁻¹(5/8)。

Inverse trig functions are usually accessed by pressing ‘shift’ or ‘2nd’ along with the sin, cos, or tan key. Your calculator will display the angle in degrees if in degree mode.

反三角函数通常通过按下 ‘shift’ 或 ‘2nd’ 键配合 sin、cos 或 tan 键来使用。若处于度模式，计算器将显示以度为单位的角。

Remember to round your answer appropriately, usually to one decimal place or three significant figures, unless the question states otherwise.

记得适当保留答案的小数位数，通常精确到一位小数或三位有效数字，除非题目另有说明。

4. Exact Trigonometric Values for Special Angles | 特殊角三角函数精确值

For OCR GCSE, you are expected to know the exact values of sin, cos, and tan for 0°, 30°, 45°, 60°, and 90° without a calculator. These can be derived from isosceles right triangles and equilateral triangles.

OCR GCSE 要求你不借助计算器记住 0°、30°、45°、60° 和 90° 时正弦、余弦和正切的精确值。这些值可以从等腰直角三角形和等边三角形中推导。

To rationalise denominators, 1/√3 becomes √3/3 and 1/√2 becomes √2/2, but the exam usually accepts either form. The pattern for sine: 0, ½, 1/√2, √3/2, 1.

Published by TutorHao | GCSE Mathematics Revision Series | aleveler.com

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