📚 Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
Trigonometry is one of the most practical topics in IGCSE Mathematics. It allows you to find unknown side lengths and angles in right-angled triangles using three key ratios. This article explains each ratio clearly, shows step-by-step methods, and highlights common mistakes to avoid.
三角学是IGCSE数学中最实用的主题之一。它利用三个关键比值帮助你在直角三角形中求出未知边长和角度。本文将清晰讲解每个比值，演示分步方法，并提醒你避免常见错误。
1. Right-Angled Triangle Basics | 直角三角形基础
Every calculation in this section starts with labelling the triangle correctly. In a right-angled triangle, one angle is 90°. The longest side, opposite the right angle, is called the hypotenuse. For a given acute angle θ, the opposite side is across the triangle from θ, and the adjacent side is next to θ and is not the hypotenuse.
本节中的每个计算都要从正确标注三角形开始。在直角三角形中，有一个角为90°。最长的那条边称为斜边，它对着直角。对于给定的锐角θ，对边是三角形中与θ相对的边，邻边是θ旁边的边且不是斜边。
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
The three definitions above are the foundation of all IGCSE trigonometry problems. You must be able to identify hypotenuse, opposite, and adjacent for any labelled diagram before attempting a calculation.
上面三个定义是所有IGCSE三角学问题的基础。在尝试任何计算之前，你必须能够为任意标注图形识别出斜边、对边和邻边。
2. The Sine Ratio | 正弦比
The sine ratio links an angle to the opposite side and the hypotenuse. It is most useful when you know the angle and want the opposite side, or when you know the opposite side and the hypotenuse and want the angle.
正弦比将一个角与对边和斜边联系在一起。当你已知角度要求对边，或已知对边和斜边要求角度时，它最为常用。
sin θ = opposite / hypotenuse
For example, if a right-angled triangle has an angle of 30° and its hypotenuse is 10 cm, then the opposite side is given by opposite = hypotenuse × sin θ = 10 × sin 30° = 5 cm. Make sure your calculator is in degree mode whenever you work with degrees.
例如，如果一个直角三角形有一个30°角，斜边为10 cm，则对边为 对边 = 斜边 × sin θ = 10 × sin 30° = 5 cm。使用角度时，请确保计算器处于角度模式（DEG）。
3. The Cosine Ratio | 余弦比
The cosine ratio connects an angle to the adjacent side and the hypotenuse. Use cosine when the adjacent side and hypotenuse are involved in the problem.
余弦比将一个角与邻边和斜边联系起来。当问题涉及邻边和斜边时，应使用余弦。
cos θ = adjacent / hypotenuse
In a right-angled triangle, suppose the hypotenuse is 5 m and the angle is 60°. Then the adjacent side is 5 × cos 60° = 2.5 m. Remember that cosine decreases as the angle increases from 0° to 90°, while sine increases, and tangent grows without bound.
在直角三角形中，假设斜边为5 m，角度为60°，则邻边为 5 × cos 60° = 2.5 m。请记住，当角度从0°增加到90°时，余弦值在减小，正弦值在增大，而正切值会无限增大。
4. The Tangent Ratio | 正切比
The tangent ratio links the opposite side and the adjacent side. It is perfect for problems where you know two legs of the triangle, or the angle and one leg, and need the other leg.
正切比将对边和邻边联系起来。当你已知直角三角形的两条直角边，或已知一个角和一条直角边、需要求另一条直角边时，正切比非常适用。
tan θ = opposite / adjacent
For instance, if tan θ = 2, then θ = tan⁻¹(2) ≈ 63.43°. In a triangle, if the opposite side is 8 cm and the adjacent side is 4 cm, then tan θ = 8 ÷ 4 = 2, and θ is therefore about 63.43°.
例如，若 tan θ = 2，则 θ = tan⁻¹(2) ≈ 63.43°。在一个三角形中，若对边为8 cm，邻边为4 cm，则 tan θ = 8 ÷ 4 = 2，因此θ约为63.43°。
5. SOHCAHTOA: A Simple Mnemonic | SOHCAHTOA：简单记忆法
To remember the three ratios, use the mnemonic SOHCAHTOA:
为了记住这三个比值，可使用口诀 SOHCAHTOA：
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SOH: Sine = Opposite ÷ Hypotenuse
SOH：正弦 = 对边 ÷ 斜边
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CAH: Cosine = Adjacent ÷ Hypotenuse
CAH：余弦 = 邻边 ÷ 斜边
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TOA: Tangent = Opposite ÷ Adjacent
TOA：正切 = 对边 ÷ 邻边
Write SOHCAHTOA at the top of your exam page as soon as you start the trigonometry section. This one action saves time and reduces errors.
考试中一进入三角函数部分，就在试卷顶端写下SOHCAHTOA。这一个动作能节省时间并减少错误。
6. Finding a Missing Side | 求未知边
When asked to find a missing side, follow a four-step strategy. First, label the sides relative to the given angle: H, O, A. Second, identify which ratio contains the side you know and the side you want. Third, write the equation. Fourth, solve it.
当要求求未知边时，请遵循四步策略。首先，相对于已知角标出各边：H（斜边）、O（对边）、A（邻边）。其次，找出同时包含已知边和所求边的比值。第三，列出方程。第四，求解。
Example: A right-angled triangle has a 40° angle and a hypotenuse of 12 cm. Find the opposite side.
示例：一个直角三角形有一个40°角，斜边为12 cm，求对边。
Solution: Because O and H are involved, use sine. sin 40° = x ÷ 12, so x = 12 × sin 40° ≈ 7.71 cm.
解：由于涉及对边O和斜边H，使用正弦。sin 40° = x ÷ 12，所以 x = 12 × sin 40° ≈ 7.71 cm。
x = 12 × sin 40° ≈ 7.71 cm
7. Finding a Missing Angle | 求未知角
To find an angle, you need the inverse trigonometric functions: sin⁻¹, cos⁻¹, and tan⁻¹. For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. On most calculators, press the shift key before the sine button.
要求角度，你需要使用反三角函数：sin⁻¹、cos⁻¹和tan⁻¹。例如，若 sin θ = 0.5，则 θ = sin⁻¹(0.5) = 30°。大多数计算器上需先按Shift键再按正弦键。
Example: In a triangle, the opposite side is 5 cm and the hypotenuse is 13 cm. Find θ.
示例：在三角形中，对边为5 cm，斜边为13 cm，求θ。
Solution: sin θ = 5 ÷ 13, so θ = sin⁻¹(5 ÷ 13) ≈ 22.62°.
解：sin θ = 5 ÷ 13，因此 θ = sin⁻¹(5 ÷ 13) ≈ 22.62°。
θ = sin⁻¹(5 ÷ 13) ≈ 22.62°
8. Bearings and Angles of Elevation | 方位角与仰角
Trigonometry is used to solve real-life problems such as bearings and angles of elevation. Bearings are measured clockwise from north and are written as three-digit angles. For a right-angled triangle formed by bearings, choose the appropriate ratio and solve.
三角学用于解决方位角和仰角等现实问题。方位角从正北方向顺时针测量，写为三位数角度。对于由方位角形成的直角三角形，选择合适的比值并求解。
For example, a ladder leaning against a wall makes an angle of 70° with the ground. If the wall is 4 m high, the length of the ladder is the hypotenuse. Since sin θ = opposite ÷ hypotenuse, we have hypotenuse = opposite ÷ sin θ = 4 ÷ sin 70° ≈ 4.26 m.
例如，一架梯子斜靠墙壁，与地面成70°角。若墙高4 m，梯子的长度就是斜边。因为 sin θ = 对边 ÷ 斜边，所以 斜边 = 对边 ÷ sin θ = 4 ÷ sin 70° ≈ 4.26 m。
9. Common Mistakes to Avoid | 常见错误及避免方法
Several errors repeatedly cost students marks. Using the wrong ratio, forgetting to set the calculator to degrees, mixing up opposite and adjacent, and using the wrong inverse function are the most common.
有几个错误反复让学生失分。使用错误的比值、忘记将计算器设为角度模式、混淆对边和邻边，以及使用错误的反三角函数都是最常见的。
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Always label the sides before writing the ratio.
在写出比值前务必定好各边名称。
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Check the calculator mode: DEG, not RAD.
检查计算器模式为度（DEG），而不是弧度（RAD）。
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When finding an angle, use the inverse function, not the standard one.
求角度时使用反函数，而不是普通函数。
10. Quick Revision Questions | 快速复习题
Try these questions on your own before checking the answers.
请先自己尝试以下问题，再查看答案。
| Question | 问题 | Answer | 答案 |
|---|---|
| 1. In a right-angled triangle, the hypotenuse is 20 cm and the angle is 35°. Find the opposite side. 1. 在直角三角形中，斜边为20 cm，角度为35°，求对边。 |
20 × sin 35° ≈ 11.47 cm |
| 2. The opposite side is 6 cm and the adjacent side is 8 cm. Find θ. 2. 对边为6 cm，邻边为8 cm，求θ。 |
tan θ = 6 ÷ 8,
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