IGCSE Mathematics: Trigonometric Ratios and Identities | 三角函数与恒等式

📚 IGCSE Mathematics: Trigonometric Ratios and Identities | 三角函数与恒等式

Trigonometry forms a core part of the IGCSE Mathematics syllabus. It connects geometric intuition with algebraic reasoning and appears in every exam paper, from simple right-angled triangle problems to graph transformations and worded applications. A clear understanding of the definitions, key values, and identities is essential for top marks.

三角学是 IGCSE 数学课程的核心内容。它将几何直觉与代数推理紧密相连，在每份试卷中都会出现——从简单的直角三角形问题，到函数图像变换和实际应用题。清楚理解定义、特殊值和恒等式，是获得高分的关键。

1. The Basic Trigonometric Ratios | 基本三角函数比

In a right-angled triangle, the three primary ratios are defined in terms of the sides:

在直角三角形中，三个基本比值由三边关系定义：

sin θ = opposite / hypotenuse (正弦)
cos θ = adjacent / hypotenuse (余弦)
tan θ = opposite / adjacent (正切)

Many students remember these with the acronym SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

很多同学用口诀“SOH CAH TOA”记忆：正弦对边比斜边，余弦邻边比斜边，正切对边比邻边。

The value of each ratio depends only on the angle θ, not on the size of the triangle. This makes the ratios useful for solving triangles of any scale.

每个比值的取值只取决于角 θ，而与三角形大小无关。因此这些比值可用于求解任意尺度的三角形。

2. Exact Values for Key Angles | 特殊角的精确值

IGCSE candidates are expected to know exact trigonometric values for angles 0°, 30°, 45°, 60° and 90° without a calculator.

IGCSE 考生需要熟记 0°、30°、45°、60° 和 90° 的三角函数精确值，并能在不使用计算器的情况下写出答案。

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	undefined

These values come from the 30-60-90 and 45-45-90 special triangles. Learning them fully removes the need to sketch triangles in the exam.

这些值来源于 30°-60°-90° 和 45°-45°-90° 特殊三角形。熟记它们后，考试时无需再现场画三角形推导。

3. The Unit Circle and Quadrants | 单位圆与象限

For angles greater than 90°, the trigonometric ratios are best understood using the unit circle, where the x-coordinate is cos θ and the y-coordinate is sin θ.

对于大于 90° 的角，利用单位圆理解三角比最为直观：单位圆上点的横坐标是 cos θ，纵坐标是 sin θ。

In the four quadrants, the signs of the ratios change:

在四个象限中，各三角比的正负会发生变化：

Quadrant I (0°–90°): all ratios are positive | 第一象限（0°–90°）：全部为正
Quadrant II (90°–180°): only sin is positive | 第二象限（90°–180°）：仅 sin 为正
Quadrant III (180°–270°): only tan is positive | 第三象限（180°–270°）：仅 tan 为正
Quadrant IV (270°–360°): only cos is positive | 第四象限（270°–360°）：仅 cos 为正

A useful mnemonic is “All Silly Turtles Cry” – Across the quadrants, the first letters are A, S, T, C, standing for All, Sin, Tan, Cos.

一个有用口诀是“All Silly Turtles Cry”——按象限顺序首字母为 A、S、T、C，分别代表 All、Sin、Tan、Cos。

4. Graphs of Sine, Cosine and Tangent | 正弦、余弦与正切图像

The graph of y = sin θ is a wave that starts at 0, reaches a maximum of 1 at 90°, then falls to -1 at 270°.

y = sin θ 的图像是波浪线，从 0 开始，在 90° 时达到最大值 1，在 270° 时降到 -1。

The graph of y = cos θ has the same shape as sine but is shifted left by 90°: it starts at 1, falls to -1 at 180°, and returns to 1 at 360°.

y = cos θ 的图像与正弦图像形状相同，但向左平移了 90°：从 1 开始，在 180° 时降到 -1，在 360° 时回到 1。

The graph of y = tan θ is very different: it repeats every 180° and has vertical asymptotes at 90°, 270°, etc., where it is undefined.

y = tan θ 的图像则完全不同：每 180° 重复一次，在 90°、270° 等处存在垂直渐近线，因为此时正切无定义。

Period of sin and cos: 360° | 正弦和余弦周期：360°
Period of tan: 180° | 正切周期：180°

When solving trigonometric equations, the graph helps you find all possible solutions within a given interval.

在解三角方程时，图像能帮助你找到给定区间内的所有可能解。

5. Solving Basic Trigonometric Equations | 解基本三角方程

Consider the equation sin θ = 0.5 for 0° ≤ θ ≤ 360°. The acute reference angle is 30°, because sin 30° = 0.5.

以方程 sin θ = 0.5（0° ≤ θ ≤ 360°）为例，其锐角参考角为 30°，因为 sin 30° = 0.5。

Since sine is positive in Quadrants I and II, the solutions are θ = 30° and θ = 180° – 30° = 150°.

因为正弦在第一、第二象限为正，所以解为 θ = 30° 和 θ = 180° – 30° = 150°。

For tan θ = 1, the reference angle is 45°, and tangent is positive in Quadrants I and III, so θ = 45° and θ = 180° + 45° = 225°.

对于 tan θ = 1，参考角为 45°，正切在第一、第三象限为正，因此 θ = 45° 和 θ = 180° + 45° = 225°。

For cos θ = 0.5, cosine is positive in Quadrants I and IV, so θ = 60° and θ = 360° – 60° = 300°.

对于 cos θ = 0.5，余弦在第一、第四象限为正，因此 θ = 60° 和 θ = 360° – 60° = 300°。

Always sketch the graph or use the CAST rule to avoid missing solutions.

始终画出图像或使用 CAST 法则，以避免漏解。

6. Trigonometric Identities | 三角恒等式

The two most important identities at IGCSE level are:

IGCSE 阶段最重要的两个恒等式是：

sin²θ + cos²θ = 1

tan θ = sin θ / cos θ

The first identity follows from the Pythagorean theorem applied to the unit circle. It allows you to write sin θ in terms of cos θ, or vice versa.

第一个恒等式可由单位圆上的勾股定理推出。它允许你用 cos θ 表示 sin θ，或者反过来表示。

For example, if sin θ = 3/5 and θ is acute, then cos²θ = 1 – (3/5)² = 16/25, so cos θ = 4/5.

例如，若 sin θ = 3/5 且 θ 为锐角，则 cos²θ = 1 – (3/5)² = 16/25，所以 cos θ = 4/5。

The second identity is used to simplify rational expressions involving sine and cosine, and to prove other identities.

第二个恒等式常用于化简包含正弦和余弦的分式，也用于证明其他恒等式。

7. Sine Rule | 正弦定理

For any triangle, not only right-angled triangles, the sine rule states:

对于任意三角形（不仅限于直角三角形），正弦定理给出：

a / sin A = b / sin B = c / sin C

Here, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

其中，边 a 对角 A，边 b 对角 B，边 c 对角 C。

The sine rule is used when you know:

正弦定理适用于已知以下条件的情况：

Two angles and one side (AAS or ASA) – use it to find the remaining side.
Two sides and one non-included angle (SSA) – use it to find an unknown angle.

两角一边（AAS 或 ASA）——用它求另一条边。
两边及其中一边的对角（SSA）——用它求未知角。

Be careful with the SSA case: it can give two possible triangles (the ambiguous case).

注意 SSA 情形可能产生两个解（即“双解”情况）。

8. Cosine Rule | 余弦定理

The cosine rule relates the three sides of a triangle to one of its angles:

余弦定理将三角形的三条边与其一个角联系起来：

a² = b² + c² – 2bc cos A

Use the cosine rule when you know:

余弦定理适用于已知以下条件的情况：

Two sides and the included angle (SAS) – use it to find the third side.
Three sides (SSS) – rearrange the formula to find any angle.

两边及其夹角（SAS）——用它求第三边。
三条边（SSS）——变形公式求任意角。

To find an angle, rearrange to:

若要求角，可将公式变形为：

cos A = (b² + c² – a²) / (2bc)

The negative value of cos A indicates an obtuse angle, which the sine rule would miss.

如果 cos A 为负，说明角 A 是钝角，这在使用正弦定理求角时容易被忽略。

9. Area of a Triangle | 三角形面积公式

The standard area formula for a triangle is extended to a non-right-angled version using sine:

三角形面积公式可以借助正弦推广到非直角三角形：

Area = ½ ab sin C

This formula is ideal when you know two sides and the included angle. For example, if a = 6 cm, b = 8 cm, and C = 30°, then:

这个公式在已知两边及其夹角时最为好用。例如，若 a = 6 cm，b = 8 cm，C = 30°，则：

Area = ½ × 6 × 8 × sin 30° = ½ × 48 × 0.5 = 12 cm²

Remember that sin 30° = 0.5, so the calculation is quick. Many exam questions give the area and ask for a missing angle or side.

请记住 sin 30° = 0.5，这样计算会很快。许多考题会给出面积，反过来求未知角或边。

10. Common Pitfalls and Exam Tips | 常见错误与考试技巧

One frequent mistake is using the calculator in the wrong mode: always check whether the question requires degrees or radians. IGCSE Paper 2 and 4 usually use degrees unless otherwise stated.

一个常见错误是计算器模式设错：始终确认题目要求的是角度制还是弧度制。IGCSE Paper 2 和 Paper 4 若无特别说明，通常使用角度制。

Another mistake is forgetting that sin θ = sin(180° – θ). When solving equations, always write down the reference angle and then apply the quadrant rule.

另一个常见错误是忘记 sin θ = sin(180° – θ)。解方程时，先写出参考角，再应用象限规则。

When using the cosine rule to find an angle, ensure you substitute the sides in the correct positions. The largest side is always opposite the largest angle.

使用余弦定理求角时，务必正确代入各边位置。最大边总是对着最大角。

Finally, draw a clear diagram for every worded problem. Many marks are lost simply because a student misread a diagram or labelled the wrong triangle.

最后，每道文字题都要画清晰的示意图。很多失分仅仅是因为看错图或标错了三角形。

11. Worked Example | 综合例题

Triangle PQR has PQ = 9 cm, PR = 7 cm, and angle QPR = 50°. Find QR.

三角形 PQR 中，PQ = 9 cm，PR = 7 cm，∠QPR = 50°。求 QR。

Here we know two sides and the included angle, so we use the cosine rule. Let QR = p, PQ = r = 9, PR = q = 7, and angle QPR = P = 50°.

已知两边及其夹角，因此使用余弦定理。设 QR = p，PQ = r = 9，PR = q = 7，∠QPR = P = 50°。

p² = q² + r² – 2qr cos P

p² = 7² + 9² – 2 × 7 × 9 × cos 50°

p² = 49 + 81 – 126 × cos 50°

Using cos 50° ≈ 0.6428, we get g² = 130 – 81.0 = 49.0, so QR = p ≈ √49 = 7.0 cm.

取 cos 50° ≈ 0.6428，则 p² = 130 – 81.0 = 49.0，所以 QR ≈ √49 = 7.0 cm。

Always include units in your final answer, and give a decimal or exact form as requested by the question.

最终答案要带上单位，并按题目要求保留小数形式或精确值。

12. Practice Questions | 练习建议

To strengthen your trigonometry skills, attempt a mix of the following question types:

为了强化三角学技能，建议混合练习以下题型：

Find an unknown side in a right-angled triangle using SOH CAH TOA.
Find an unknown angle in a non-right-angled triangle using the sine rule.
Find the third side of a triangle using the cosine rule.
Calculate the area of a triangle using ½ ab sin C.
Solve equations such as 2 cos x + 1 = 0 in the range 0° ≤ x ≤ 360°.
Prove or simplify expressions using sin²θ + cos²θ = 1.

用 SOH CAH TOA 求直角三角形的未知边。
用正弦定理求非直角三角形的未知角。
用余弦定理求三角形的第三边。
用 ½ ab sin C 计算三角形面积。
在 0° ≤ x ≤ 360° 范围内解方程，例如 2 cos x + 1 = 0。
利用 sin²θ + cos²θ = 1 证明或化简表达式。

By practising these examples without a calculator for exact values, and with a calculator for longer calculations, you will build both speed and accuracy.

先用不含计算器的方式练习特殊角的精确值，再用计算器进行复杂计算，这样既能提速，又能提高准确率。

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IGCSE Mathematics: Trigonometric Ratios and Identities | 三角函数与恒等式