GCSE Maths: Trigonometry Key Concepts | 三角函数考点精讲

📚 GCSE Maths: Trigonometry Key Concepts | 三角函数考点精讲

Trigonometry is a core topic in GCSE Mathematics, linking angles and side lengths in triangles. Mastering the sine, cosine and tangent ratios, the sine and cosine rules, and the graphs of trigonometric functions will give you the tools to solve a wide range of problems. This revision guide covers every essential concept, from right‑angled triangle basics to the sine rule and the area formula, with clear examples and exact value tables.

三角学是 GCSE 数学的核心主题，它将三角形中的角度与边长联系起来。掌握正弦、余弦和正切比，正弦定理和余弦定理，以及三角函数图像，将使你有能力解决各种问题。这份复习指南涵盖了所有基本概念，从直角三角形基础到正弦定理和面积公式，配有清晰的例子和精确值表格。

1. Labelling a Right‑Angled Triangle | 直角三角形标记法

In any right‑angled triangle, the longest side, opposite the right angle, is called the hypotenuse. When we focus on one of the two acute angles, the side opposite that angle is the opposite, and the side next to the angle (that is not the hypotenuse) is the adjacent.

在任何直角三角形中，最长的那条边，就是直角所对的边，称为斜边。当我们关注两个锐角中的一个时，这个角所对的边称为对边，与这个角相邻的边（不是斜边的那条）称为邻边。

For example, in triangle ABC with the right angle at C, side AB is the hypotenuse. If we label angle A, then the side opposite A is BC (the opposite), and the side next to A is AC (the adjacent).

例如，在直角三角形 ABC 中，直角在 C 点，边 AB 就是斜边。如果我们标记角 A，那么 A 的对边就是 BC（对边），与 A 相邻的边就是 AC（邻边）。

These labels are essential because the three trigonometric ratios – sine, cosine and tangent – are defined by the sides relative to a chosen angle.

这些标记至关重要，因为三个基本的三角比——正弦、余弦和正切——都是根据相对于所选角的边来定义的。

2. SOHCAHTOA: The Three Ratios | SOHCAHTOA：三个基本比

The mnemonic SOHCAHTOA helps you remember the definitions:

助记词 SOHCAHTOA 可以帮助你记住这些定义：

SOH: sin θ = Opposite / Hypotenuse

SOH: sin θ = 对边 / 斜边

CAH: cos θ = Adjacent / Hypotenuse

CAH: cos θ = 邻边 / 斜边

TOA: tan θ = Opposite / Adjacent

TOA: tan θ = 对边 / 邻边

Here θ (theta) is one of the acute angles in the right‑angled triangle. These ratios are constant for a given angle, no matter how large the triangle is, because all such triangles are similar.

这里的 θ（theta）是直角三角形中的一个锐角。对于一个给定的角，这些比值是固定的，与三角形的大小无关，因为所有这样的三角形都是相似的。

You must be able to identify which ratio to use based on the sides you know or need to find. SOHCAHTOA is the first tool you will apply in most GCSE trigonometry questions.

你必须能够根据已知或需要求的边，来判断使用哪个比。在大多数 GCSE 三角学题目中，SOHCAHTOA 是你首先会应用的工具。

3. Finding Missing Sides | 求未知边长

When you know one angle (other than the right angle) and one side of a right‑angled triangle, you can find any other side using the appropriate trigonometric ratio. Set up an equation from SOHCAHTOA, substitute the known values and solve for the unknown.

当你已知直角三角形的一个锐角和一条边时，就可以用相应的三角比求出任何其他边。从 SOHCAHTOA 出发列出方程，代入已知值，然后解出未知数。

For example, suppose the hypotenuse is 12 cm and the angle is 35°. To find the opposite side, use sin: sin 35° = opposite / 12, so opposite = 12 × sin 35°. Using a calculator gives about 6.88 cm.

例如，假设斜边长为 12 cm，角度为 35°。要求对边，使用 sin：sin 35° = 对边 / 12，因此对边 = 12 × sin 35°。用计算器算出的结果约为 6.88 cm。

Similarly, if you know the adjacent side and need the hypotenuse, use cos. If you know the opposite and adjacent, use tan. Always check that your calculator is set to degree mode – a very common mistake at GCSE.

类似地，如果已知邻边而需要求斜边，就用 cos。如果已知对边和邻边，就用 tan。务必检查计算器是否设置为“度”模式——这是 GCSE 考试中一个非常常见的错误。

4. Finding Missing Angles | 求未知角度

To find an acute angle when two sides are known, use the inverse trigonometric functions: sin⁻¹, cos⁻¹ and tan⁻¹ (often accessed by pressing shift then sin, cos or tan). Choose the ratio that involves the two known sides.

要在已知两条边时求锐角，需要使用反三角函数：sin⁻¹、cos⁻¹ 和 tan⁻¹（通常通过按 shift 键然后按 sin、cos 或 tan 来调用）。选择那两个已知边所对应的比。

For instance, if the opposite side is 5 cm and the adjacent side is 8 cm, then tan θ = 5/8, so θ = tan⁻¹(5/8) ≈ 32.0°. Once again, ensure the answer is given in degrees, usually to one decimal place or as specified.

例如，如果对边长 5 cm，邻边长 8 cm，那么 tan θ = 5/8，因此 θ = tan⁻¹(5/8) ≈ 32.0°。同样地，确保答案用度表示，通常保留一位小数或按题目要求处理。

When the hypotenuse and the opposite are known, use sin⁻¹; when the hypotenuse and the adjacent are known, use cos⁻¹. Writing down the SOHCAHTOA equation first will help you avoid mix‑ups.

当已知斜边和对边时，使用 sin⁻¹；当已知斜边和邻边时，使用 cos⁻¹。先把 SOHCAHTOA 的式子写下来，可以避免混淆。

5. Exact Values of Trigonometric Ratios | 特殊角的精确值

For certain key angles, you are expected to know the exact values of sin, cos and tan without a calculator. These angles are 0°, 30°, 45°, 60° and 90°. The table below summarises the values; all fractions are given in their simplest form and denominators are rationalised where appropriate.

对于某些关键角度，你不会用到计算器就需要知道 sin、cos 和 tan 的精确值。这些角度是 0°、30°、45°、60° 和 90°。下表总结了这些值；所有分数都已化为最简形式，分母在适当处已有理化。

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

These values often appear in non‑calculator papers. It helps to remember the patterns: for 0°, 30°, 45°, 60°, 90°, the sin values follow √0/2, √1/2, √2/2, √3/2, √4/2, and cos values are the reverse order.

这些值经常出现在非计算器试卷中。记住规律会很有帮助：对于 0°、30°、45°、60°、90°，sin 值依次为 √0/2、√1/2、√2/2、√3/2、√4/2，而 cos 值则顺序相反。

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

In many word problems, trigonometry is used to find heights or distances. The angle of elevation is the angle above the horizontal when you look up at an object. The angle of depression is the angle below the horizontal when you look down.

在许多应用题中，三角学被用来求高度或距离。仰角是当你向上看一个物体时，视线与水平线之间的夹角。俯角是当你向下看时的角度。

These angles are always measured from the horizontal, and the observer’s line of sight forms the hypotenuse of a right‑angled triangle. A common trick is that the angle of depression from A to B equals the angle of elevation from B to A, because they are alternate angles.

这些角度总是从水平线开始测量，观察者的视线则构成直角三角形的斜边。有一个常用技巧：从 A 看 B 的俯角等于从 B 看 A 的仰角，因为它们互为内错角。

Sketch a clear diagram with all given information, then label the sides relative to the angle you are using, and apply SOHCAHTOA. For example, given the distance from a building and the elevation angle to the top, you can find the building’s height using tan.

画一个清晰的示意图，标上所有已知信息，然后相对于所使用的角标出各边，再应用 SOHCAHTOA。例如，已知离一栋建筑的距离以及到楼顶的仰角，你就可以用 tan 求出建筑的高度。

7. The Sine Rule | 正弦定理

The sine rule works for any triangle, not just right‑angled ones. It states that the ratio of a side length to the sine of its opposite angle is constant:

正弦定理适用于任何三角形，而不仅仅是直角三角形。它表明，边长与其对角的正弦之比是一个常数：

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

Alternatively, it can be written as sin A / a = sin B / b = sin C / c. Use the sine rule when you know either two angles and one side (AAS or ASA) or two sides and a non‑included angle (SSA) – but be careful with the ambiguous case.

它也可以写为 sin A / a = sin B / b = sin C / c。当已知两角一边（AAS 或 ASA）或两边及一个非夹角（SSA）时，使用正弦定理——但要注意二义性情况。

For example, if angle A = 40°, angle B = 60° and side a = 8 cm, then b = (sin 60° / sin 40°) × 8 cm. Solve step by step and always use the unrounded values until the final answer.

例如，如果角 A = 40°，角 B = 60°，边 a = 8 cm，那么 b = (sin 60° / sin 40°) × 8 cm。一步一步地求解，并且在最终答案之前一直使用未舍入的值。

The sine rule is also used in finding the area of a triangle, and it links neatly with the circumcircle of the triangle (where a / sin A = 2R).

正弦定理也用于求三角形的面积，并且它与三角形的外接圆有密切联系（其中 a / sin A = 2R）。

8. The Cosine Rule | 余弦定理

The cosine rule is another powerful tool for non‑right‑angled triangles. It connects all three sides and one angle:

余弦定理是处理非直角三角形的另一个有力工具。它将三条边与一个角联系起来：

a² = b² + c² − 2bc cos A

To find an angle when all three sides are known, rearrange it as cos A = (b² + c² − a²) / (2bc). Use the cosine rule when you know two sides and the included angle (SAS) and want the third side, or when you know all three sides (SSS) and need an angle.

当已知三边求角时，可将公式变形为 cos A = (b² + c² − a²) / (2bc)。当已知两边及其夹角（SAS）并求第三边时，或当已知三边（SSS）并需要求一个角时，使用余弦定理。

For instance, if b = 7 cm, c = 10 cm and angle A = 50°, then a² = 7² + 10² − 2×7×10×cos 50°. Work out the right side and then take the square root. Always write the formula down before substituting numbers.

例如，若 b = 7 cm，c = 10 cm，角 A = 50°，则 a² = 7² + 10² − 2×7×10×cos 50°。先计算右边的值，再开平方。在代入数字之前，务必先把公式写下来。

The cosine rule is essentially a generalisation of Pythagoras’ theorem: when angle A = 90°, cos 90° = 0 and the formula reduces to a² = b² + c².

余弦定理本质上是勾股定理的推广：当角 A = 90° 时，cos 90° = 0，公式就变为 a² = b² + c²。

9. Area of a Triangle: ½ ab sin C | 三角形面积公式：½ ab sin C

When you know two sides and the angle between them, the area of any triangle can be found without the perpendicular height:

当你知道两条边及其夹角时，无需知道垂直高也可以求出任意三角形的面积：

Area = ½ ab sin C

Here a and b are the lengths of two sides, and C is the included angle. The formula works because a sin C gives the perpendicular height if side b is taken as the base.

这里 a 和 b 是两条边的边长，C 是它们的夹角。这个公式之所以成立，是因为如果以 b 为底，a sin C 就是垂直高。

For example, a triangle has sides a = 6 cm, b = 8 cm and the included angle C = 30°. Then area = ½ × 6 × 8 × sin 30° = 24 × ½ = 12 cm². Note that you must always use the included angle – the angle between the two known sides.

例如，三角形有边长 a = 6 cm，b = 8 cm，夹角 C = 30°。那么面积 = ½ × 6 × 8 × sin 30° = 24 × ½ = 12 cm²。注意，你必须使用这两条已知边之间的夹角。

This formula is extremely useful in multi‑step problems where you first need to find an angle using the sine or cosine rule, then compute the area. It also reinforces the idea that the maximum area for two given sides occurs when the angle between them is 90°.

这个公式在多步骤问题中非常有用，例如先需要用正弦或余弦定理求出一个角，再计算面积。它也强化了一个概念：对于给定的两条边，当夹角为 90° 时面积最大。

10. Graphs of Trigonometric Functions | 三角函数图像

Understanding the graphs of y = sin x, y = cos x and y = tan x is essential for solving equations and modelling periodic behaviour. For GCSE Higher, you need to know their shapes, intercepts, turning points and asymptotes for 0° ≤ x ≤ 360°.

理解 y = sin x、y = cos x 和 y = tan x 的图像对于解方程和建立周期模型至关重要。在 GCSE Higher 级别，你需要知道它们在 0° ≤ x ≤ 360° 范围内的形状、截距、转折点和渐近线。

The sine graph starts at 0, rises to a maximum of 1 at 90°, returns to 0 at 180°, reaches −1 at 270° and finishes at 0 at 360°. Its period is 360° and amplitude is 1. The cosine graph is identical in shape but shifted 90° to the left: it starts at 1, hits 0 at 90°, −1 at 180°, 0 at 270° and 1 at 360°.

正弦图像从 0 开始，在 90° 时上升到最大值 1，在 180° 回到 0，在 270° 达到 −1，在 360° 回到 0。它的周期是 360°，振幅是 1。余弦图像形状相同，但向左平移了 90°：起点为 1，90° 时为 0，180° 时为 −1，270° 时为 0，360° 时回到 1。

The tangent graph looks very different: it has vertical asymptotes at 90° and 270°, where tan x is undefined. It passes through the origin, increases sharply near 90°, then appears from negative infinity just after 90°, crossing the x‑axis again at 180° and repeating the pattern with period 180°.

正切图像看起来很不一样：它在 90° 和 270° 处有垂直渐近线，此处 tan x 无定义。图像通过原点，在接近 90° 时急剧上升，然后从负无穷大在 90° 之后重新出现，在 180° 处再次穿过 x 轴，并以 180° 为周期重复这一模式。

Recognising these graphs allows you to estimate solutions to trigonometric equations and to see why, for instance, sin θ = sin(180° − θ). Practise sketching them quickly and labelling the key points.

熟悉这些图像能让你估计三角方程的解，并理解为什么例如 sin θ = sin(180° − θ) 之类的恒等式成立。练习快速画出草图并标注关键点。

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GCSE Maths: Trigonometry Key Concepts | 三角函数考点精讲