📚 Mastering Trigonometric Ratios | 掌握三角比
Trigonometry in right-angled triangles is a core IGCSE Mathematics topic. It allows you to find unknown sides and angles using sine, cosine and tangent ratios. This guide explains the key definitions, the SOHCAHTOA method, exact values, and exam strategies.
直角三角形中的三角学是 IGCSE 数学的核心主题。它使你能够利用正弦、余弦和正切比求出未知边和角。本指南讲解关键定义、SOHCAHTOA 方法、精确值以及应试策略。
1. Right-Angled Triangles and Labelling | 直角三角形与边的标记
Every trigonometric ratio begins with a right-angled triangle. The three sides are labelled relative to a chosen acute angle θ: the hypotenuse is the longest side, opposite the right angle; the opposite side faces θ; the adjacent side is next to θ but is not the hypotenuse.
每个三角比都从直角三角形开始。相对于所选锐角 θ，三条边被标记为：斜边是最长边，对着直角；对边正对 θ；邻边紧邻 θ 但不是斜边。
Good labelling is essential before applying sine, cosine or tangent. If you choose the other acute angle, the labels for opposite and adjacent swap. Always draw and label the triangle first in any problem.
在应用正弦、余弦或正切之前，正确标记边非常重要。如果选择另一个锐角，对边和邻边的标记会互换。在任何问题中都要先画图并标记三角形。
2. Pythagoras' Theorem as a Foundation | 勾股定理作为基础
Pythagoras' ypotenuse. This is used to find a missing side when two sides are known, and it underpins many trigonometric relationships.
勾股定理指出，在直角三角形中，a² + b² = c²，其中 c 是斜边。当已知两条边时，它用于求缺失的边，并且它是许多三角关系的基础。
For example, if the two shorter sides are 3 cm and 4 cm, the hypotenuse is √3² + 4² = √25 = 5 cm. This 3-4-5 triangle often appears in IGCSE questions and forms a useful check.
例如，如果两条较短边为 3 cm 和 4 cm，则斜边为 √3² + 4² = √25 = 5 cm。这个 3-4-5 三角形经常出现在 IGCSE 题目中，是有用的检验。
3. Introducing Sine, Cosine and Tangent | 引入正弦、余弦和正切
θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent.
对于直角三角形中的锐角 θ，三个主要的三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
These ratios depend only on the angle θ, not on the size of the triangle. This makes them powerful tools for scaling lengths and solving real-world measurement problems.
这些比只取决于角 θ，而与三角形的大小无关。这使得它们成为按比例缩放长度和解决现实测量问题的有力工具。
4. The SOHCAHTOA Memory Aid | SOHCAHTOA 记忆法
potenuse; TOA gives Tan = Opposite over Adjacent. Many learners find it helps to say it as "so-cah-toe-ah".
SOHCAHTOA 是一个简单的记忆法：SOH 表示 Sin = 对边 / 斜边；CAH 表示 Cos = 邻边 / 斜边；TOA 表示 Tan = 对边 / 邻边。许多学生发现把它读作 "so-cah-toe-ah" 很有帮助。
Before selecting a ratio, identify which side is unknown and which sides are known. This tells you whether to use sine, cosine or tangent. If only sides are involved, Pythagoras may be faster.
在选择比之前，先确定哪条边未知、哪条边已知。这告诉你该用正弦、余弦还是正切。如果只涉及边，勾股定理可能更快。
5. Finding a Missing Side | 求缺失的边
uation such as sin 35° = x / 12, then multiply both sides by the denominator to isolate x.
要求缺失的边，应选择能将已知角和已知边与未知边联系起来的比。建立例如 sin 35° = x / 12 的方程，然后将两边乘以分母以解出 x。
x = 12 × sin 35° ≈ 6.88 m. Always write the substituted formula before using a calculator.
例如，如果一架 12 m 长的梯子与地面成 35° 角，则达到的高度为 x = 12 × sin 35° ≈ 6.88 m。在使用计算器之前，应先写出代入数值后的公式。
6. Finding a Missing Angle | 求缺失的角
6, then θ = sin⁻¹(0.6) ≈ 36.9°. On most calculators, press shift and then sin, cos or tan.
如果已知两条边并要求角度，就使用反三角函数。例如，若 sin θ = 0.6，则 θ = sin⁻¹(0.6) ≈ 36.9°。在大多数计算器上，先按 shift 再按 sin、cos 或 tan。
Always check your answer is sensible: the missing angle must be less than 90° in a right-angled triangle. If you get a negative angle or an error, you may have labelled the sides incorrectly.
始终检查答案是否合理：直角三角形中的缺失角必须小于 90°。如果你得到负角或错误，可能说明你把边标记错了。
7. Exact Values of Special Angles | 特殊角的精确值
IGCSE candidates are often asked for exact values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°. These come from special right-angled triangles: the 45° isosceles triangle and the 30°-60° triangle.
IGCSE 考生经常被要求给出 0°、30°、45°、60° 和 90° 时 sin、cos 和 tan 的精确值。这些值来自两个特殊直角三角形：45° 等腰直角三角形和 30°-60° 三角形。
The key exact values are
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