📚 Mastering IGCSE Trigonometry: Ratios, Graphs and Problem Solving | 掌握 IGCSE 三角函数：比值、图像与应用题
Trigonometry is a high-yield topic in IGCSE Mathematics. It appears in questions on right-angled triangles, bearings, graphs, and non-right-angled triangles. This guide explains the key formulas, exact values, and problem-solving steps in a clear way. Work through each section and sketch the diagrams yourself.
三角函数是 IGCSE 数学中的高频考点，出现在直角三角形、方位角、图像以及非直角三角形等问题中。本指南清楚地讲解关键公式、精确值和解题步骤。请逐节学习并自己画出相应图形。
1. What Trigonometry Is | 什么是三角函数
Trigonometry is the study of the relationships between the angles and the side lengths of a triangle. At IGCSE level you will meet it in right-angled triangles, in non-right-angled triangles, in bearings questions, in graph work and in three-dimensional problems.
三角函数研究三角形中角度与边长之间的关系。在 IGCSE 阶段，你会在直角三角形、非直角三角形、方位角问题、图像题以及三维问题中遇到它。
Before you touch any formula, always identify the type of triangle and label the sides relative to the angle you are working with. In a right-angled triangle, the side opposite the right angle is the hypotenuse, and it is always the longest side. The side across from the given acute angle θ is the opposite, and the side next to θ but not the hypotenuse is the adjacent.
在套用任何公式之前，务必先判断三角形类型，并根据你所使用的角标注各边。在直角三角形中，直角所对的边是斜边，它总是最长的一条边；已知锐角 θ 对面的边是对边；与 θ 相邻但不构成斜边的边是邻边。
The labels opposite and adjacent only make sense relative to a chosen angle. If you switch to the other acute angle in the same triangle, the opposite and adjacent sides swap over while the hypotenuse stays the same.
"对边"和"邻边"只有相对某个选定的角才有意义。如果你换成同一个三角形中的另一个锐角，对边和邻边会互换，而斜边保持不变。
2. Right-Angled Triangle Ratios | 直角三角形中的比值
The three basic trigonometric ratios relate an acute angle to two sides of a right-angled triangle. Each ratio is a fraction, so it is a pure number with no units.
三个基本三角函数比值将锐角与直角三角形的两条边联系起来。每个比值都是一个分数，因此是没有单位的纯数。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
A common memory aid is "SOH CAH TOA": Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
常用的记忆口诀是 "SOH CAH TOA"：正弦等于对边除以斜边，余弦等于邻边除以斜边，正切等于对边除以邻边。
For a given acute angle, all three ratios are fixed by the shape of the triangle rather than its size. This is exactly why similar triangles share the same trigonometric values for the same angle.
对于给定的锐角，三个比值由三角形的形状决定，而与大小无关。这正是相似三角形中相同角的三角函数值相等的原因。
3. Exact Values for Key Angles | 特殊角的精确值
For the non-calculator paper, you must know the exact values of sine, cosine and tangent at 0°, 30°, 45°, 60° and 90°. These values can be derived from an equilateral triangle of side 2 cut in half and from a square of side 1, but memorising them saves time.
对于非计算器试卷，你必须掌握 0°、30°、45°、60° 和 90° 时正弦、余弦和正切的精确值。这些值可以从边长为 2 的等边三角形对折以及边长为 1 的正方形中推导，但记住它们可以节省时间。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 = √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Note that tan 90° is undefined, because the adjacent side would have length zero and division by zero is not allowed. Also 1/√3 is normally written as √3/3 after rationalising the denominator.
注意 tan 90° 无定义，因为此时邻边长度为零，而除以零是不允许的。另外 1/√3 通常在分母有理化后写成 √3/3。
Two patterns make the table easy to recall. Reading across the sine row gives √0/2, √1/2, √2/2, √3/2 and √4/2, which simplify to 0, 1/2, √2/2, √3/2 and 1. The cosine row is the same list read backwards.
有两个规律可以让这张表更容易记住。正弦一行依次是 √0/2、√1/2、√2/2、√3/2、√4/2，化简后即 0、1/2、√2/2、√3/2、1。余弦一行就是把这份数列倒过来读。
4. Pythagoras and Trigonometric Identities | 勾股定理与三角恒等式
Pythagoras' theorem states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides, where c is the hypotenuse.
勾股定理指出，在直角三角形中，斜边的平方等于另外两条边的平方和，其中 c 是斜边。
a² + b² = c²
Combining Pythagoras with SOH CAH TOA produces two identities that appear in IGCSE extension papers. They hold for every angle θ, not just acute ones.
把勾股定理与 SOH CAH TOA 结合，可以得到两个出现在 IGCSE 进阶试卷中的恒等式。它们对任意角 θ 都成立，而不仅限于锐角。
sin² θ + cos² θ = 1 tan θ = sin θ ÷ cos θ
These identities let you find cos θ when you only know sin θ, or simplify an expression such as sin θ ÷ cos θ, without reaching for a calculator.
这些恒等式让你在只知道 sin θ 时求出 cos θ，或者在不使用计算器的情况下化简像 sin θ ÷ cos θ 这样的表达式。
For example, if sin θ = 3/5 and θ is acute, then cos θ = √1 − 9/25 = √16/25 = 4/5, and tan θ = 3/4.
例如，若 sin θ = 3/5 且 θ 为锐角，则 cos θ = √1 − 9/25 = √16/25 = 4/5，且 tan θ = 3/4。
