GCSE AQA Maths: Trigonometry Key Points Revision | GCSE AQA 数学：三角函数考点精讲

📚 GCSE AQA Maths: Trigonometry Key Points Revision | GCSE AQA 数学：三角函数考点精讲

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. In GCSE AQA Maths, it is applied to solve problems involving right-angled triangles and non-right-angled triangles. A solid understanding of trigonometric ratios, exact values, graphs, and rules is essential for success in the exam.

三角学是研究三角形边与角之间关系的数学分支。在 GCSE AQA 数学中，它被用于解决直角三角形和非直角三角形的问题。扎实掌握三角比、精确值、图像和定理是考试成功的关键。

1. Introduction to Trigonometry | 三角函数简介

Trigonometry deals with the measurement of triangles. It provides a set of tools that link angles to side lengths, allowing us to calculate unknown dimensions when some information is given. The word comes from the Greek trigōnon (triangle) and metron (measure). In GCSE, we focus on practical applications in two-dimensional and some three-dimensional problems.

三角学研究三角形的度量，提供了一套将角度与边长联系起来的工具，使我们能够在已知部分信息时计算未知尺寸。这个词源于希腊语 trigōnon（三角形）和 metron（度量）。在 GCSE 中，我们侧重于二维和部分三维问题中的实际应用。

The core concepts include right-angled triangle ratios (sine, cosine, tangent), the unit circle definitions, graphs of trigonometric functions, and rules for non-right-angled triangles such as the sine rule and cosine rule. AQA also tests the ability to recall exact trigonometric values for key angles without a calculator.

核心概念包括直角三角形比（正弦、余弦、正切）、单位圆定义、三角函数图像以及适用于非直角三角形的正弦定理和余弦定理。AQA 还考查在不用计算器的情况下回忆关键角的精确三角函数值的能力。

2. Right-Angled Triangles and SOHCAHTOA | 直角三角形与 SOHCAHTOA

In a right-angled triangle, the three primary trigonometric ratios are defined relative to one of the acute angles. The hypotenuse is the longest side, opposite the right angle. The opposite and adjacent sides depend on the angle you are considering. The mnemonic SOHCAHTOA helps remember the relationships:

在直角三角形中，三个主要三角比是针对其中一个锐角定义的。斜边是最长边，正对直角。对边和邻边取决于所考虑的角。记忆口诀 SOHCAHTOA 帮助记住这些关系：

SOH: Sin = Opposite / Hypotenuse | 正弦 = 对边 / 斜边
CAH: Cos = Adjacent / Hypotenuse | 余弦 = 邻边 / 斜边
TOA: Tan = Opposite / Adjacent | 正切 = 对边 / 邻边

When labelling the triangle, always identify the right angle first, then label the sides relative to the angle you know or want to find. This will ensure you use the correct ratio. The same triangle can be labelled differently when focusing on the other acute angle.

标记三角形时，务必先找出直角，然后根据已知或需要求的角标记各边。这能确保使用正确的比。当聚焦于另一个锐角时，同一个三角形的标记会有所不同。

3. Calculating Sides and Angles | 计算边长与角度

To find a missing side, choose the appropriate ratio based on the known angle and the sides you are dealing with. Substitute the known values and rearrange the equation. For example, if you know an angle and the hypotenuse and need the opposite side, use sin: opposite = hypotenuse × sin(angle).

要求缺失的边长，根据已知角和涉及的边选择合适的比。代入已知值并重新排列方程。例如，如果知道一个角和斜边，需要求对边，则使用正弦：对边 = 斜边 × sin(角)。

To find a missing angle, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). On most calculators these are accessed by pressing ‘shift’ and then the sin, cos or tan button. Make sure your calculator is set to degree mode. The presence of a small ‘D’ or ‘DEG’ on the display confirms this.

要求缺失的角度，使用反三角函数（sin⁻¹，cos⁻¹，tan⁻¹）。在大多数计算器上，通过按 ‘shift’ 再按 sin、cos 或 tan 键来操作。确保计算器处于度数模式。显示屏上有小 ‘D’ 或 ‘DEG’ 字样即可确认。

Always show steps clearly: write the equation, substitute, solve, and round appropriately. AQA often requires answers to be given to a specified number of significant figures or decimal places. Use your calculator efficiently, but show working out to avoid losing method marks.

务必清晰展示步骤：写出方程、代入、求解并适当取整。AQA 经常要求答案给出指定有效数字或小数位数。高效使用计算器，但要展示解题过程以免失去方法分。

4. Exact Trigonometric Values | 特殊角的精确三角函数值

GCSE AQA expects you to know the exact values of sin, cos and tan for 0°, 30°, 45°, 60° and 90° without using a calculator. These values can be derived from special right triangles: the isosceles right triangle (45°-45°-90°) with sides in ratio 1:1:√2, and the half-equilateral triangle (30°-60°-90°) with sides in ratio 1:√3:2.

GCSE AQA 要求你在不使用计算器的情况下记住 0°、30°、45°、60° 和 90° 时 sin、cos 和 tan 的精确值。这些值可以从特殊的直角三角形推导出来：等腰直角三角形（45°-45°-90°）的边长比例为 1:1:√2，以及半等边三角形（30°-60°-90°）的边长比例为 1:√3:2。

Angle θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3 / 2	1 / √3
45°	1 / √2 (or √2/2)	1 / √2 (or √2/2)	1
60°	√3 / 2	1/2	√3
90°	1	0	undefined

Practice writing these values from memory and recognise patterns: sin increases from 0 to 1, cos decreases from 1 to 0. The values for sin 30° and cos 60° are the same, as are sin 45° and cos 45°, which reflects the complementary angle relationship sin θ = cos (90° – θ).

练习默写这些值并识别规律：sin 从 0 增加到 1，cos 从 1 减小到 0。sin 30° 和 cos 60° 的值相同，sin 45° 和 cos 45° 的值相同，这反映了互余角关系 sin θ = cos (90° – θ)。

5. Trigonometric Graphs | 三角函数图像

The graphs of y = sin x, y = cos x and y = tan x are part of the AQA syllabus. You must be able to sketch them, identify their key features, and solve equations graphically. All three functions are periodic, meaning their shapes repeat at regular intervals.

函数 y = sin x，y = cos x 和 y = tan x 的图像是 AQA 考纲的一部分。你必须能够画出它们的草图，识别关键特征，并用图解法解方程。这三个函数都是周期性的，意味着它们的形状以固定间隔重复。

y = sin x: A wave that starts at 0, reaches 1 at 90°, returns to 0 at 180°, goes to -1 at 270°, and returns to 0 at 360°. The period is 360°. It is an odd function with symmetry about the origin.

y = sin x： 波形从 0 开始，在 90° 时达 1，180° 时回到 0，270° 时到 -1，360° 时回到 0。周期为 360°。它是奇函数，关于原点对称。

y = cos x: Starts at 1, drops to 0 at 90°, -1 at 180°, 0 at 270°, back to 1 at 360°. Period 360°. It is an even function, symmetric about the y-axis.

y = cos x： 从 1 开始，90° 时降为 0，180° 时到 -1，270° 时回到 0，360° 时回到 1。周期 360°。它是偶函数，关于 y 轴对称。

y = tan x: Has asymptotes at 90°, 270°, etc. Crosses the x-axis at 0°, 180°, 360°. Period 180°. It is unbounded and repeats a curved S-shape between asymptotes.

y = tan x： 在 90°、270° 等处有渐近线。在 0°、180°、360° 处穿过 x 轴。周期 180°。它是无界的，在渐近线之间重复弯曲的 S 形。

Common exam tasks include reading solutions for sin x = k from the graph and using symmetry to find all solutions within a given range, e.g. 0° ≤ x ≤ 360°. For sin, if one solution is α, the other is 180° – α. For cos, the other is 360° – α. For tan, add 180° to α.

常见的考题包括从图像上读取 sin x = k 的解，并利用对称性求出给定范围（如 0° ≤ x ≤ 360°）内的所有解。对于 sin，若一个解为 α，另一个是 180° – α。对于 cos，另一个是 360° – α。对于 tan，则将 α 加上 180°。

6. Angles of Elevation and Depression | 仰角与俯角

These angles are measured relative to the horizontal. The angle of elevation is the angle upwards from the horizontal to an object. The angle of depression is the angle downwards from the horizontal to an object. Both are commonly used in contextual problems involving height and distance.

这些角是相对于水平线测量的。仰角是从水平线向上到物体的角。俯角是从水平线向下到物体的角。两者常用于涉及高度和距离的实际问题中。

When drawing a diagram, assume the horizontal lines are parallel; this makes the angle of depression from one point equal to the angle of elevation from the opposite point (alternate angles). Use this to build a right-angled triangle and apply SOHCAHTOA. Always check that your diagram and chosen ratio match the situation.

画图时，假设各条水平线平行；这使得从一点看下去的俯角等于从另一点看回来的仰角（内错角相等）。利用这一点构造直角三角形并应用 SOHCAHTOA。务必检查图和所选比值是否与题意相符。

7. Bearings and Trigonometry | 方位角与三角函数

A bearing is a three-digit angle measured clockwise from North. Bearings are used to describe direction accurately. AQA exam questions often combine bearings with trigonometry to find distances between points or a specific bearing back to the origin.

方位角是从正北顺时针测量的三位数角度，用于精确描述方向。AQA 试题经常将方位角与三角函数结合，求两点之间的距离或返回原点的特定方位角。

To solve such problems, draw a clear North line at each point. The bearing from A to B is measured at A. If you need the return bearing from B to A, it may be the forward bearing plus or minus 180°, adjusted to remain within 000° to 360°. Identify the right-angled triangles, use alternate angles to find missing angles, and apply trigonometric ratios.

解决此类问题时，在每一点处画一条清晰的北方向线。从 A 到 B 的方位角是在 A 处测量的。如果需要从 B 返回 A 的方位角，它可能是正向方位角加减 180°，并通过调整保持在 000° 到 360° 范围内。找出直角三角形，利用内错角求出缺失的角，并应用三角比。

8. Sine Rule | 正弦定理

The sine rule is used for non-right-angled triangles when you know either two angles and any side (AAS or ASA) or two sides and a non-included angle (SSA). The formula relates sides and the sines of opposite angles:

正弦定理用于非直角三角形，当已知两角及任一边（AAS 或 ASA）或两边及一个非夹角（SSA）时使用。公式将边长与其对角的正弦联系起来：

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

To find a missing side, use the two relevant parts. For a missing angle, rearrange to sin A = (a × sin B) / b and then use the inverse sine. Be mindful of the ambiguous case when given SSA: there may be two possible angles (acute and obtuse) because sin θ = sin (180° – θ). In GCSE, normally only one valid triangle will fit the given data, but check the context.

要求缺失的边长，使用相关的两部分。要求缺失的角度，重新排列为 sin A = (a × sin B) / b，然后使用反正弦。当给出 SSA 时要注意模糊情况：可能有两个解（锐角和钝角），因为 sin θ = sin (180° – θ)。在 GCSE 中，通常只有一个有效的三角形符合所给数据，但需结合题意检查。

9. Cosine Rule | 余弦定理

The cosine rule is another tool for non-right-angled triangles, particularly useful when you have two sides and the included angle (SAS) or three sides (SSS). It is often seen as an extension of Pythagoras’ theorem. The most common forms are:

余弦定理是非直角三角形的另一个工具，尤其适用于已知两边及其夹角（SAS）或三边（SSS）的情况。它常被视为勾股定理的推广。最常见的形式是：

a² = b² + c² – 2bc cos A

To find an angle, rearrange to cos A = (b² + c² – a²) / (2bc). Carefully label the triangle so that the side opposite the angle you are working with matches the letter on the left-hand side. Take care with the order of operations when substituting values.

要求角时，重新排列为 cos A = (b² + c² – a²) / (2bc)。仔细标记三角形，使所处理的角的对边与等式左边的字母一致。代入数值时注意运算顺序。

10. Area of a Triangle: ½ab sin C | 三角形面积：½ab sin C

When the perpendicular height is not known, the area of any triangle can be found using two sides and the included angle:

当垂直高度未知时，任何三角形的面积可用两边及其夹角求得：

Area = ½ ab sin C

Here a and b are the lengths of two sides, and C is the angle between them. This formula is particularly useful because it avoids having to construct a perpendicular height. For right-angled triangles, it reduces to ½ × base × height when sin 90° = 1.

这里 a 和 b 是两边的长度，C 是它们之间的夹角。这个公式特别有用，因为它避免了作垂直高度。对于直角三角形，当 sin 90° = 1 时，它简化为 ½ × 底 × 高。

In exam questions, you may need to use the formula both ways: finding area or finding a missing angle/side when area is given. Always include the units in area calculations (e.g., cm²) and watch for mixed unit conversions.

在考题中，你可能需要双向运用该公式：求面积，或给定面积时求缺失的角或边。面积计算中务必写上单位（如 cm²），并注意混合单位的换算。

11. Problem Solving and Exam Tips | 综合问题与复习技巧

Trigonometry questions often involve multiple steps: drawing the diagram, identifying the correct rule, performing algebraic manipulation, and rounding appropriately. Always sketch and label the triangle clearly. If the problem includes a 3D shape, extract the 2D triangle you need and work in planes.

三角学问题常涉及多个步骤：画图、识别正确的定理、进行代数运算以及正确取整。务必清晰画出三角形并标注。如果问题包含三维形状，提取出所需的二维三角形并在平面内求解。

Common pitfalls to avoid:

Forgetting to set calculator to degree mode. | 忘记将计算器设为度数模式。
Mixing up opposite and adjacent sides. | 混淆对边和邻边。
Using the cosine rule when the sine rule is more direct. | 在正弦定理更直接时使用余弦定理。
Not checking if the answer is reasonable (e.g., a side length much larger than the hypotenuse is impossible). | 未检查答案是否合理（如边长远大于斜边是不合理的）。
Rounding too early, which can cause final answer inaccuracies. | 过早取整会导致最终答案不准确。

Practice past paper questions to become fluent in recognising the structure of each problem. When revising, create summary cards with exact values, graph sketches, and the two rules. This will sharpen your recall under exam pressure.

练习历年真题，熟练识别每道题的结构。复习时制作包含精确值、图像草图及两个定理的摘要卡片，这有助于在考试压力下快速回忆。

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GCSE AQA Maths: Trigonometry Key Points Revision | GCSE AQA 数学：三角函数考点精讲