Drawing a 3-4-5 Triangle: A Foundation for Trigonometry | 绘制3-4-5直角三角形：三角学的基石

📚 Drawing a 3-4-5 Triangle: A Foundation for Trigonometry | 绘制3-4-5直角三角形：三角学的基石

The 3-4-5 right-angled triangle is one of the simplest and most powerful tools in A-Level mathematics. It appears in trigonometry, geometry, and coordinate problems, and drawing it by hand helps students understand where sine, cosine, and tangent ratios actually come from. In this article, we explore the construction, the mathematics behind it, and how exam questions use this classic triangle.

3-4-5直角三角形是A-Level数学中最简单却最有力的工具之一。它出现在三角学、几何和坐标问题中，而亲手绘制这个三角形能帮助学生真正理解正弦、余弦和正切比值的来源。在本文中，我们将探讨它的构造、背后的数学原理，以及考试题目如何运用这个经典三角形。

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1. What Is a 3-4-5 Triangle? | 什么是3-4-5三角形？

A 3-4-5 triangle is a right-angled triangle whose three side lengths are exactly 3, 4 and 5 units. The longest side, length 5, is always the hypotenuse, which sits opposite the right angle. The two shorter sides, 3 and 4, are the legs that form the right angle.

3-4-5三角形是一个三边长度分别为3、4、5个单位的直角三角形。最长的边（长度为5）永远是斜边，它与直角相对。两条较短的边3和4则是构成直角的直角边。

The three integers form a Pythagorean triple, meaning they satisfy the relationship a² + b² = c². In this case:

这三个整数构成一组毕达哥拉斯三元组，即它们满足关系 a² + b² = c²。在本例中：

3² + 4² = 9 + 16 = 25 = 5²

This equality guarantees that a triangle with these side lengths must contain a 90° angle, by the converse of Pythagoras’ theorem. Therefore, if you only know the side lengths and not the angle, you can be certain the triangle is right-angled.

这个等式保证，由这三条边构成的三角形必然包含一个90°角，这是毕达哥拉斯定理的逆定理。因此，即使你只知道边长而不知道角度，也能确信该三角形是直角三角形。

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2. Drawing the Triangle Step by Step | 分步绘制三角形

Drawing a 3-4-5 triangle is straightforward with a ruler and a set square. Start with a horizontal base of length 4 units. At one end, draw a vertical line of length 3 units using a set square to ensure it is perpendicular. Join the two free ends with a line of length 5 units.

用直尺和三角板绘制3-4-5三角形非常简单。先画一条长度为4个单位的水平底边。在底边的一端，用三角板确保垂直，画一条长度为3个单位的竖直线。再用一条长度为5个单位的线段连接两个自由端点。

When drawn accurately, the longest side should measure exactly 5 units, which confirms the construction. You can also test this with a protractor: the angle between the base and the vertical side is 90°, and the two acute angles are approximately 36.87° and 53.13°.

当绘制准确时，最长边应正好为5个单位，这验证了构造的正确性。你也可以用量角器测试：底边与竖直边之间的角度为90°，两个锐角分别约为36.87°和53.13°。

For A-Level problems, you do not always need to draw the triangle physically — visualising a sketch with the 3, 4, 5 labels is enough to extract the trigonometric ratios. Making a habit of sketching labelled triangles is one of the most effective revision strategies for trigonometry questions.

对于A-Level题目，你不必总是实际画出三角形——只需在草图中标注3、4、5三边，就足以提取三角函数比值。养成画出带标记三角形的习惯，是应对三角函数题最有效的复习策略之一。

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3. Labelling the Sides | 标记各边

Before finding trigonometric ratios, we must label the sides correctly. Choose one of the two acute angles as the reference angle, often denoted θ. The side opposite this angle is the “opposite” side, the side next to it (but not the hypotenuse) is the “adjacent” side.

在求三角函数比值之前，我们必须正确标记各边。选择两个锐角之一作为参考角，通常记为θ。与该角相对的边是”对边”，与它相邻（但不是斜边）的边是”邻边”。

• If θ is at the corner between the sides of length 4 and 5, then the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5.

  如果θ位于长度为4和5的两边之间的角，则对边为3，邻边为4，斜边为5。

• If θ is at the corner between the sides of length 3 and 5, then the opposite side is 4, the adjacent side is 3, and the hypotenuse is still 5.

  如果θ位于长度为3和5的两边之间的角，则对边为4，邻边为3，斜边仍为5。

This flexibility means the same triangle gives us two complementary sets of ratios. If you label the angle incorrectly, every ratio you write down will be wrong, so always annotate your sketch before doing any calculation.

这种灵活性意味着同一个三角形为我们提供两组互为余角的比值。如果角度标记错误，你写出的所有比值都会出错，所以在进行任何计算之前，务必在草图上做好标注。

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4. Sine, Cosine and Tangent Ratios | 正弦、余弦和正切比值

Using the definitions of trigonometric ratios, we can write down exact fractions for each acute angle in the 3-4-5 triangle. Remember the mnemonic SOH-CAH-TOA:

利用三角比的定义，我们可以为3-4-5三角形中的每个锐角写出精确分数。记住口诀SOH-CAH-TOA：

sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent

sin θ = 对边 ÷ 斜边，cos θ = 邻边 ÷ 斜边，tan θ = 对边 ÷ 邻边

For the angle where the opposite side is 3 and adjacent is 4, we obtain the three fundamental ratios:

对于对边为3、邻边为4的角，我们得到三个基本比值：

sin θ = 3⁄5, cos θ = 4⁄5, tan θ = 3⁄4

For the other acute angle, which we may call φ, the opposite and adjacent sides swap, so the ratios also swap:

对于另一个锐角，我们称之为φ，对边与邻边互换，因此比值也随之互换：

sin φ = 4⁄5, cos φ = 3⁄5, tan φ = 4⁄3

These are exact and rational, making the 3-4-5 triangle extremely convenient for non-calculator exam questions. The full set of values is summarised below:

这些比值是精确的有理数，使得3-4-5三角形在不能使用计算器的考试题中极为方便。全部数值汇总如下：

Angle | 角度	sin	cos	tan
θ (opposite 3 | 对边为3)	3⁄5	4⁄5	3⁄4
φ (opposite 4 | 对边为4)	4⁄5	3⁄5	4⁄3

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5. Finding the Acute Angles | 求锐角大小

To find the actual angle measures, we apply inverse trigonometric functions. Using the ratio sin θ = 3⁄5:

为了求实际角度大小，我们应用反三角函数。利用比值 sin θ = 3⁄5：

θ = arcsin(3⁄5) ≈ 36.87°

The other acute angle is therefore 90° − 36.87° = 53.13°, which we can also obtain from arcsin(4⁄5) or arctan(4⁄3).

因此另一个锐角为 90° − 36.87° = 53.13°，我们也可以由 arcsin(4⁄5) 或 arctan(4⁄3) 得到它。

These are not “nice” angles like 30°, 45° or 60°, but their ratios are clean fractions. In AQA exam questions, you are often told to round to 1 decimal place, or you work with the ratios directly without

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