IGCSE AQA Maths: Trigonometry Key Points Review | IGCSE AQA 数学：三角函数考点精讲

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

Trigonometry is a fundamental branch of mathematics that deals with the relationships between angles and sides in triangles. In the AQA IGCSE Mathematics syllabus, trigonometry appears in both Foundation and Higher tiers, covering right-angled triangles, exact values, bearings, and for Higher students, the sine and cosine rules, graphs of trigonometric functions, and 3D applications. Mastering these topics is essential for achieving a high score on your exam.

三角学是数学的一个基本分支，研究三角形中角度与边之间的关系。在 AQA IGCSE 数学教学大纲中，三角函数既出现于基础层也出现于进阶层，涵盖直角三角形、精确三角值、方位角，而对于进阶层还包括正弦与余弦定理、三角函数图像以及三维应用。掌握这些内容是在考试中取得高分的关键。

1. Right-Angled Triangle Trigonometry (SOH CAH TOA) | 直角三角形三角比（SOH CAH TOA）

In a right-angled triangle, the three primary trigonometric ratios are defined relative to a given acute angle θ:

在直角三角形中，三个基本三角比相对于一个给定锐角 θ 定义如下：

Sine (sin θ) = opposite side / hypotenuse. Remember ‘SOH’: Sine is Opposite over Hypotenuse.

正弦 (sin θ) = 对边 / 斜边。记住 ‘SOH’：正弦是对边除以斜边。

Cosine (cos θ) = adjacent side / hypotenuse. ‘CAH’: Cosine is Adjacent over Hypotenuse.

余弦 (cos θ) = 邻边 / 斜边。’CAH’：余弦是邻边除以斜边。

Tangent (tan θ) = opposite side / adjacent side. ‘TOA’: Tangent is Opposite over Adjacent.

正切 (tan θ) = 对边 / 邻边。’TOA’：正切是对边除以邻边。

Always identify the hypotenuse (longest side, opposite the right angle), the opposite side (across from angle θ), and the adjacent side (next to θ, not the hypotenuse) before applying these ratios.

在应用这些比值之前，一定要先确认斜边（最长边，对应直角）、对边（角 θ 对面）和邻边（紧邻 θ，不是斜边）。

2. Finding Missing Sides and Angles | 求未知边与角

To find a missing side, choose the ratio that links the known angle and the known side to the unknown side. Set up an equation and solve. For example, if you know angle θ and the hypotenuse h, and need the opposite side o: sin θ = o / h → o = h × sin θ.

要求未知边，选择能够将已知角、已知边与未知边联系起来的三角比。列出方程并求解。例如，已知角 θ 和斜边 h，要求对边 o：sin θ = o / h → o = h × sin θ。

To find a missing angle when two sides are known, use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹). If sin θ = opposite / hypotenuse, then θ = sin⁻¹(opposite / hypotenuse). Ensure your calculator is in degree mode.

当已知两边求未知角时，使用反三角函数 (sin⁻¹, cos⁻¹, tan⁻¹)。若 sin θ = 对边 / 斜边，则 θ = sin⁻¹(对边 / 斜边)。确保计算器处于角度模式。

Always round your final answer to a suitable degree of accuracy, typically 1 decimal place for angles and 3 significant figures for side lengths.

最终答案通常按照适当的精度四舍五入，角度保留 1 位小数，边长保留 3 位有效数字。

3. Exact Trigonometric Values for Key Angles | 特殊角的精确三角值

You are expected to memorise the exact values of sine, cosine, and tangent for 30°, 45°, and 60°. These appear frequently in non-calculator papers.

你需要记住 30°、45° 和 60° 的正弦、余弦和正切的精确值。这些值经常出现在非计算器试卷中。

Exact trigonometric values / 精确三角值

Angle θ	sin θ	cos θ	tan θ
30°	½	√3 / 2	1 / √3
45°	√2 / 2	√2 / 2	1
60°	√3 / 2	½	√3

Notice the symmetry: sin 30° = cos 60°, and sin 60° = cos 30°. For tan, tan 45° = 1, and tan 30° and tan 60° are reciprocals.

注意对称性：sin 30° = cos 60°，sin 60° = cos 30°。对于正切，tan 45° = 1，而 tan 30° 与 tan 60° 互为倒数。

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

The angle of elevation is the angle measured upwards from the horizontal to a line of sight looking at an object above. The angle of depression is the angle measured downwards from the horizontal to a line of sight looking at an object below.

仰角是从水平线向上到看向上方物体的视线的夹角。俯角是从水平线向下到看向下方物体的视线的夹角。

These angles are always measured from the horizontal and are equal in magnitude if the lines of sight are parallel. Drawing a clear right-angled triangle connecting the observer, the object, and the horizontal is key to solving such problems.

这些角总是从水平线量起，如果视线平行，它们的大小相等。画出连接观察者、物体和水平线的清晰直角三角形是解决这类问题的关键。

For example, from a point 50 m away from the base of a tower, the angle of elevation to the top is 30°. The height of the tower can be found using tan 30° = height / 50 → height = 50 × (1/√3) meters.

例如，从距离塔底 50 m 的一点测得塔顶仰角为 30°。塔的高度可以用 tan 30° = 高度 / 50 求得 → 高度 = 50 × (1/√3) 米。

5. Bearings and Trigonometry | 方位角与三角学

Bearings are used to indicate direction. They are measured clockwise from North and are always written as three-digit numbers (e.g., 045°, 210°). Bearings are often combined with trigonometry to find distances and positions.

方位角用于表示方向。它们从正北顺时针测量，并总是用三位数字书写（如 045°、210°）。方位角常与三角学结合以求解距离与位置。

When solving bearing problems, draw a diagram with a north line at each point. Look for right-angled triangles formed by north–south and east–west lines. Use alternate interior angles to find angles within the triangle, then apply SOH CAH TOA.

解决方位角问题时，需在每个点画出指北线。寻找由南北线和东西线形成的直角三角形。利用内错角找出三角形内的角，然后应用 SOH CAH TOA。

For instance, a ship sails 10 km on a bearing of 070°. Its eastward displacement is 10 × sin 70°, northward is 10 × cos 70°.

例如，一艘船沿方位角 070° 航行 10 km，其向东位移为 10 × sin 70°，向北位移为 10 × cos 70°。

6. The Sine Rule | 正弦定理

For any triangle (not just right-angled), the sine rule states:

对于任意三角形（不仅直角三角形），正弦定理指出：

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

or equivalently: sin A / a = sin B / b = sin C / c

或等价地：sin A / a = sin B / b = sin C / c

The sine rule is used when you know: (i) two angles and one side (AAS or ASA) to find another side, or (ii) two sides and a non-included angle (SSA) to find an angle.

正弦定理用于以下情况：(i) 已知两角一边 (AAS 或 ASA) 求另一边，或 (ii) 已知两边及非夹角 (SSA) 求一角。

Be cautious with the SSA case; it can sometimes produce two possible triangles (the ambiguous case). Always check whether the angle you find is acute or obtuse based on the triangle’s information.

注意 SSA 情况，有时可能产生两个可能的三角形（含糊情况）。务必根据三角形信息核查求出的角是锐角还是钝角。

7. The Cosine Rule | 余弦定理

The cosine rule applies to any triangle and is particularly useful when the sine rule is not applicable:

余弦定理适用于任意三角形，在正弦定理不适用时尤其有用：

a² = b² + c² − 2bc cos A

This can be rearranged to find an angle:

该公式可变形以求角：

cos A = (b² + c² − a²) / (2bc)

Use the cosine rule when you know: (i) three sides (SSS) to find any angle, or (ii) two sides and the included angle (SAS) to find the remaining side.

在以下情况使用余弦定理：(i) 已知三边 (SSS) 求任意角，或 (ii) 已知两边及其夹角 (SAS) 求第三边。

Remember to keep careful track of which side corresponds to which angle. Labeling triangles consistently (side a opposite angle A, etc.) helps avoid mistakes.

牢记哪条边对应哪个角。统一标注三角形（边 a 对角 A 等）有助于避免错误。

8. Area of a Triangle (½ab sin C) | 三角形面积公式 (½ab sin C)

For a non-right-angled triangle, the area is given by:

对于非直角三角形，面积可由下式求得：

Area = ½ a b sin C

where a and b are two sides, and C is the angle between them.

其中 a 与 b 是两条边，C 是它们之间的夹角。

If the angle C is 90°, sin C = 1, and the formula reduces to the familiar area = ½ × base × height. This is a powerful tool for problems where the perpendicular height is not given directly.

如果角 C 为 90°，sin C = 1，公式退化为熟悉的面积 = ½ × 底 × 高。当题目未直接给出垂直高度时，这是一个强有力的工具。

Always ensure the angle used is the one between the two chosen sides. You may need to use sine or cosine rules first to find a required angle.

务必确保所用角是所选两边的夹角。可能需要先用正弦或余弦定理求出所需角度。

9. Graphs of sin x, cos x, tan x and Solving Equations | sin x、cos x、tan x 的图像与解方程

The graphs of y = sin x, y = cos x, and y = tan x for 0° ≤ x ≤ 360° are essential. Key features:

在 0° ≤ x ≤ 360° 的区间内，y = sin x、y = cos x 和 y = tan x 的图像至关重要。主要特征：

y = sin x: Starts at 0, rises to 1 at 90°, falls back to 0 at 180°, goes to −1 at 270°, returns to 0 at 360°. Maximum 1, minimum −1, period 360°.

y = sin x：从 0 开始，到 90° 升至 1，180° 回到 0，270° 降至 −1，360° 回到 0。最大值 1，最小值 −1，周期 360°。

y = cos x: Starts at 1 at 0°, falls to 0 at 90°, −1 at 180°, 0 at 270°, back to 1 at 360°. Also has period 360°.

y = cos x：从 0° 的 1 开始，90° 降到 0，180° 到 −1，270° 到 0，360° 回到 1。周期也为 360°。

y = tan x: Has asymptotes at 90° and 270°. It repeats every 180°, passing through 0 at 0°, 180°, 360°.

y = tan x：在 90° 和 270° 有渐近线。每 180° 重复一次，在 0°、180°、360° 过零点。

To solve an equation like sin x = 0.5 for 0° ≤ x ≤ 360°, use the graph or symmetry: x = sin⁻¹(0.5) = 30° is the acute solution. The other solution is 180° − 30° = 150°, because sin x = sin(180° − x). Always check the range.

要解方程如 sin x = 0.5，0° ≤ x ≤ 360°，利用图像或对称性求得：x = sin⁻¹(0.5) = 30° 是锐角解，另一解为 180° − 30° = 150°，因为 sin x = sin(180° − x)。务必检查求解范围。

10. Trigonometry in Three Dimensions | 三维三角学

3D trigonometry extends right-angled triangle skills to solids like cuboids, pyramids, and prisms. The challenge is to identify the correct right-angled triangle within the 3D structure.

三维三角学将直角三角形的技巧扩展到长方体、棱锥和棱柱等立体图形。关键在于在三维结构中准确找出直角三角形。

Common examples include finding the angle between a space diagonal and a face diagonal, or the angle between a sloping edge and the base. Always sketch the 2D triangle separately, labeling all known lengths clearly.

常见例子包括求空间对角线与其所在面的对角线之间的夹角，或一条侧棱与底面间的夹角。务必单独画出二维三角形图，并清楚标注所有已知边长。

Work step by step: use Pythagoras’ theorem first if necessary to find missing lengths in the base or faces, then apply SOH CAH TOA to calculate the required angle.

循序渐进：必要时先用勾股定理求出底面或某个面内的未知边长，然后应用 SOH CAH TOA 计算所求角度。

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IGCSE AQA Maths: Trigonometry Key Points Review | IGCSE AQA 数学：三角函数 考点精讲