📚 Mastering Trigonometry for IGCSE CCEA Mathematics | IGCSE CCEA 数学：三角函数 考点精讲
Trigonometry is a branch of mathematics that explores the relationships between the angles and side lengths of triangles. In the IGCSE CCEA Mathematics syllabus, this topic is fundamental for both the calculator and non‑calculator papers. You will need to understand the three primary trigonometric ratios, how to use them to solve right‑angled triangles, and how to extend these ideas to the sine rule and cosine rule for any triangle. This article covers all the key ideas, from the basic definitions to graph sketching and practical applications, helping you build confidence step by step.
三角函数是研究三角形边长与角度之间关系的数学分支。在 IGCSE CCEA 数学大纲中，这个主题是计算器与非计算器试卷的重要基础。你需要掌握三种基本的三角比，会利用它们解直角三角形，并能扩展到任意三角形的正弦定理与余弦定理。本文将从基本定义一直讲解到图像绘制与实际应用，帮助你一步步建立信心。
1. The Three Trigonometric Ratios | 三种基本三角比
In a right‑angled triangle, the ratios of the sides relative to one of the acute angles are called sine, cosine and tangent. For an angle θ, we define sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent. The position of the opposite and adjacent sides depends on which acute angle you are referring to, so always label your triangle carefully.
在直角三角形中，与某个锐角相关的边长之比分别称为正弦、余弦和正切。对于角 θ，我们定义 sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。对边和邻边是相对于你正在使用的锐角而言的，因此一定要仔细标记三角形。
It is useful to memorise the acronym SOH CAH TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. This simple phrase can help you quickly set up equations when the triangle is right‑angled.
记住口诀 SOH CAH TOA 会很有用：Sin = 对/斜，Cos = 邻/斜，Tan = 对/邻。这个简单的口诀能帮助你在直角三角形中快速列出方程。
2. Finding Sides and Angles in Right‑Angled Triangles | 解直角三角形求边与角
If you know one acute angle and one side, you can find the other sides by choosing the appropriate ratio. For example, given angle A and the hypotenuse, the opposite side = hypotenuse × sin A, and the adjacent side = hypotenuse × cos A. You can also find an acute angle when two sides are known by using the inverse trigonometric functions: θ = sin⁻¹(opposite/hypotenuse), θ = cos⁻¹(adjacent/hypotenuse), or θ = tan⁻¹(opposite/adjacent).
如果你知道一个锐角和一条边，就可以选择合适的三角比求其他边长。例如，已知角 A 和斜边，对边 = 斜边 × sin A，邻边 = 斜边 × cos A。当已知两条边时，可以使用反三角函数求锐角：θ = sin⁻¹(对边/斜边)，θ = cos⁻¹(邻边/斜边) 或 θ = tan⁻¹(对边/邻边)。
Remember to set your calculator to degree mode when dealing with angles in degrees. A common mistake is to leave it in radian mode, which produces completely different numbers. CCEA questions will nearly always use degrees unless specified otherwise.
处理角度时务必把计算器设置为度数模式。一个常见错误是把它留在弧度模式，这样得到的结果会完全不同。CCEA 试题除非特别说明，几乎都使用度作单位。
3. Exact Trigonometric Values for Key Angles | 特殊角的精确三角值
The CCEA specification expects you to know exact values for sin, cos and tan at 0°, 30°, 45°, 60° and 90°. You can derive these from two standard triangles: a right‑angled isosceles triangle with acute angles 45°–45° (sides 1, 1, √2) and an equilateral triangle split into two 30°–60°–90° triangles (sides 1, √3, 2). These exact values are often tested without a calculator.
CCEA 大纲要求你记住 0°、30°、45°、60° 和 90° 的正弦、余弦和正切的精确值。你可以通过两个标准三角形来推导：一个是等边直角三角形（45°–45°，边长为 1、1、√2），另一个是由等边三角形分出的 30°–60°–90° 三角形（边长为 1、√3、2）。这些精确值常在无计算器题中考查。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | ½ | √3 |
| 90° | 1 | 0 | undefined |
4. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is the angle measured upwards from the horizontal to an object above. An angle of depression is measured downwards from the horizontal to an object below. These angles are always measured relative to the horizontal line, not the vertical. Problems often involve two right‑angled triangles sharing a common vertical line, such as a person looking at the top and bottom of a building from a distance.
仰角是从水平线向上观察物体时的角度。俯角是从水平线向下观察物体时的角度。这些角总是相对于水平线测量，而不是垂直线。典型问题常涉及两个直角三角形共用一条垂直线，例如一个人从远处看建筑物的顶端和底部。
Draw a clear diagram and label all known lengths and angles. Then identify the right‑angled triangle that contains the required side or angle, and apply SOH CAH TOA. Sometimes you need to use two different triangles and subtract one distance from another to find a height or a horizontal distance.
画一个清晰的草图，标注所有已知长度和角度。然后找出包含所求边长或角度的直角三角形，应用 SOH CAH TOA。有时你需要利用两个不同的三角形，用一个距离减去另一个距离来求高度或水平距离。
5. The Sine Rule | 正弦定理
The sine rule applies to any triangle, not just right‑angled ones. It states that a / sin A = b / sin B = c / sin C, where a, b, c are side lengths and A, B, C are the angles opposite those sides. Equivalently, sin A / a = sin B / b = sin C / c is also correct and often easier to use when finding an angle.
正弦定理适用于任意三角形，而不仅仅是直角三角形。它指出 a / sin A = b / sin B = c / sin C，其中 a、b、c 是边长，A、B、C 分别是这些边所对的角。同样，sin A / a = sin B / b = sin C / c 的写法也是正确的，且在求角时往往更方便。
Use the sine rule when you know two angles and one side (AAS or ASA) or two sides and a non‑included angle (SSA). When using SSA, watch out for the ambiguous case: there may be two possible triangles because the unknown angle could be acute or obtuse. In CCEA exams you are expected to recognise this possibility when the given angle is acute and the side opposite it is shorter than the other given side.
当已知两角一边（AAS 或 ASA），或已知两边及一个非夹角（SSA）时，使用正弦定理。在使用 SSA 时，需要注意模糊情况：由于未知角可能是锐角也可能是钝角，可能存在两个符合条件的三角形。CCEA 考试要求你识别这种可能性，具体条件是已知角为锐角且它所对的边比另一已知边短。
6. The Cosine Rule | 余弦定理
The cosine rule links the three sides of a triangle with one of its angles. It is typically written as a² = b² + c² − 2bc cos A, where a is the side opposite angle A. Rearranging gives cos A = (b² + c² − a²) / (2bc), which is used to find an angle when all three sides are known.
余弦定理将三角形的三条边与其中一个角联系起来。通常写成 a² = b² + c² − 2bc cos A，其中 a 是角 A 的对边。移项可以得到 cos A = (b² + c² − a²) / (2bc)，用于已知三边求角。
Apply the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS). In the first situation, you solve for the unknown side; in the second, you solve for one of the angles. The cosine rule is a generalisation of Pythagoras’ theorem — when A = 90°, cos A = 0 and the formula reduces to a² = b² + c².
当已知两边及夹角（SAS）或已知三边（SSS）时，应用余弦定理。第一种情况用于求第三边；第二种情况用于求一个角。余弦定理是勾股定理的推广——当 A = 90° 时，cos A = 0，公式即退化为 a² = b² + c²。
7. Area of a Triangle Using Trigonometry | 利用三角函数求三角形面积
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 included angle between them. This formula is especially useful when you do not know the perpendicular height, which is often the case in non‑right‑angled triangles.
任何三角形的面积都可以用公式 面积 = ½ ab sin C 来求，其中 a 和 b 是两条边，C 是它们之间的夹角。当不知道垂直高度时（在非直角三角形中常见），这个公式非常有用。
Remember to use the same angle that sits between the two known sides. If you are given a different angle, you may need to use the sine rule first to find the required sides or angles. This formula also appears in problems involving bearings and navigation, where you often know two distances and the angle between the two directions.
记住要使用两条已知边之间的夹角。如果给出的不是这个角，你可能需要先用正弦定理求出所需的边长或角度。这个公式也会出现在方位角和航海中，此时你通常知道两个距离和两条方向线之间的夹角。
8. Graphs of sin x, cos x and tan x | sin x、cos x 和 tan x 的图像
The graphs of the three trigonometric functions are periodic and have distinct shapes. The graph of y = sin x oscillates between −1 and 1, passing through the origin with a period of 360°. The graph of y = cos x also oscillates between −1 and 1 but starts at (0,1) and has the same period. The graph of y = tan x repeats every 180° and has vertical asymptotes at x = 90°, 270°, … where the function is undefined.
这三个三角函数的图像是周期性的，并且形状各自不同。y = sin x 的图像在 −1 和 1 之间振荡，通过原点，周期为 360°。y = cos x 的图像同样在 −1 和 1 之间振荡，但从点 (0,1) 开始，周期相同。y = tan x 的图像每 180° 重复一次，在 x = 90°、270° 等处有竖直渐近线，函数在这些点无定义。
Understanding the graphs allows you to solve simple trigonometric equations like sin x = 0.5 within a given interval. By sketching the graph, you can see all solutions within 0° ≤ x ≤ 360° not just the principal value from your calculator. For example, sin x = 0.5 gives x = 30° and x = 150°; cos x = 0.5 gives x = 60° and x = 300°.
理解这些图像能让你在给定区间内解简单的三角方程，如 sin x = 0.5。通过画草图，你可以看到 0° 至 360° 范围内的全部解，而不仅仅是计算器给出的主值。例如，sin x = 0.5 的解为 x = 30° 和 x = 150°；cos x = 0.5 的解为 x = 60° 和 x = 300°。
9. Solving Trigonometric Equations | 解三角方程
To solve an equation like sin x = k, first use your calculator to find the principal angle, then use the symmetry of the sine graph or the CAST diagram to find additional solutions in the given range. The general rules are: for sin x = k, the second solution is 180° − θ; for cos x = k, the second solution is 360° − θ; for tan x = k, add or subtract 180° to find further solutions because the period is 180°.
要解 sin x = k 这样的方程，先用计算器求出主角，然后利用正弦图像的对称性或 CAST 图求给定范围内的其他解。一般规律是：对于 sin x = k，第二个解为 180° − θ；对于 cos x = k，第二个解为 360° − θ；对于 tan x = k，加减 180° 可得其他解，因为它的周期是 180°。
If the equation involves a coefficient inside the argument, such as sin 2x = 0.5, you should adjust the range accordingly. For 0° ≤ x ≤ 360°, the range for 2x becomes 0° ≤ 2x ≤ 720°. Find all solutions for 2x and then divide by 2 to obtain the values of x. Many students forget to expand the range, which causes them to miss solutions.
如果方程内部有系数，如 sin 2x = 0.5，你应当相应地调整区间。对于 0° ≤ x ≤ 360°，2x 的范围变为 0° ≤ 2x ≤ 720°。先找出 2x 的所有解，再除以 2 得到 x 的值。很多学生忘记扩展范围，导致漏解。
10. Bearings and Trigonometry | 方位角与三角学
Bearings are used to describe direction, measured clockwise from north, always given as three figures (e.g. 045°, 135°, 270°). Trigonometry problems involving bearings often require you to construct right‑angled triangles by drawing north‑south lines through points. The angles inside these triangles are frequently related to the bearing by subtracting from 90°, 180° or 360°.
方位角用来描述方向，从正北顺时针测量，始终用三位数字表示（例如 045°、135°、270°）。涉及方位角的三角题通常需要通过点画出南北方向线来构造直角三角形。这些三角形中的角常与方位角有关，通过从 90°、180° 或 360° 减去得到。
Draw a clean diagram with all the relevant north lines and label the distances. Use alternate angles and allied angles to find missing angles in the triangle, then apply the sine rule, cosine rule or basic trig ratios as needed. Bearings problems are an excellent test of whether you can translate a real‑world context into a mathematical model.
画一个清晰的图，标出所有相关北线和距离。利用内错角和同旁内角求出三角形中的未知角，然后根据需要应用正弦定理、余弦定理或基本三角比。方位角问题是检验你能否将实际情境转化为数学模型的好题目。
11. 3D Trigonometry | 三维三角问题
CCEA may include questions where you need to find lengths or angles in three‑dimensional shapes, such as cuboids, pyramids or prisms. The key is to identify a right‑angled triangle that lies in a plane of the 3D figure. Often you will need to use Pythagoras’ theorem first to find a diagonal length on a face, and then use trigonometry to find the angle between a line and a plane, or between two planes.
CCEA 可能会考查三维图形中的长度或角度问题，比如长方体、棱锥或棱柱。关键是找出位于三维图形某个平面内的直角三角形。你通常需要先用勾股定理求出某个面上的对角线，然后再用三角学求出直线与平面之间的夹角或两个平面之间的夹角。
The angle between a line and a plane is defined as the angle between the line and its projection onto that plane. To find it, you identify the right‑angled triangle formed by the line, its projection and the perpendicular from the top of the line to the plane. Label all known edges clearly and work step by step.
直线与平面的夹角定义为该直线与其在该平面上的投影之间的夹角。要求这个角，需要找出由直线、它的投影以及从直线顶端到平面的垂线所构成的直角三角形。清楚地标记所有已知的棱长，然后按步骤求解。
12. Common Mistakes and How to Avoid Them | 常见错误与避免方法
One of the most common errors is confusing the opposite and adjacent sides when labelling a right‑angled triangle. Always start by marking the right angle and the acute angle you are using, then identify the hypotenuse (longest side, opposite the right angle) first. The opposite side is the one facing the given acute angle, and the adjacent is the remaining side touching that angle.
最常见的一个错误是在给直角三角形做标记时混淆对边和邻边。永远先标出直角和你正在使用的锐角，然后首先确定斜边（最长的边，对着直角）。对边是面对已知锐角的边，邻边是剩下的与那个角相邻的边。
Another frequent mistake is forgetting to switch the calculator to degree mode, or rounding intermediate values too early. Always keep full calculator accuracy until the final answer, then round to the required degree of accuracy — usually three significant figures or one decimal place as directed. Also, when using the sine rule for an angle, be aware of the ambiguous case and check whether the obtuse solution is valid in the context.
另一个常见错误是忘记将计算器切换为度数模式，或者过早对中间值进行四舍五入。始终保留计算器上的全部精度直到最终答案，然后再四舍五入到要求的精确度——通常按要求保留三位有效数字或一位小数。此外，当用正弦定理求角时，要注意模糊情况，并检查钝角解在实际问题中是否成立。
Finally, always re‑read the question to confirm what you are being asked: sometimes it is the angle with the horizontal, not the vertical; sometimes you need to add or subtract heights from different triangles; and sometimes the answer must be given as a bearing, which requires a specific format.
最后，一定要重新读题，确认题目要求的是什么：有时是求与水平线的夹角而不是垂直线；有时需要将不同三角形中的高度相加或相减；还有时答案需要以方位角的形式给出，这有特定的格式要求。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导