📚 Trigonometry in Right-Angled Triangles | 直角三角形中的三角学
In IGCSE Mathematics, right-angled triangle trigonometry is one of the most frequently assessed topics. It connects angle measures with side ratios and appears in pure geometry, bearings, angles of elevation and depression, and real-life problems. Mastering sine, cosine and tangent, and knowing when to use them, gives you reliable marks across both core and extended papers.
在 IGCSE 数学中，直角三角形三角学是最常考查的主题之一。它把角度与边长比联系起来，出现在纯几何、方位角、仰角与俯角以及实际问题中。掌握正弦、余弦和正切，并知道何时使用它们，能为你在核心卷和扩展卷中稳定拿分。
1. Trigonometric Ratios | 三角比
In any right-angled triangle, label the three sides relative to the acute angle you are working with. The hypotenuse is always the longest side and lies opposite the right angle.
在任何直角三角形中，根据你正在使用的锐角来标注三条边。斜边总是最长边，并且位于直角的对面。
For a chosen angle θ, the side directly facing θ is the opposite, and the side next to θ, but not the hypotenuse, is the adjacent.
对于选定的角 θ，正对 θ 的边是对边，与 θ 相邻但不是斜边的边是邻边。
The three basic trigonometric ratios are defined as follows.
三个基本三角比的定义如下。
sin θ = opposite ÷ hypotenuse
cos θ = adjacent ÷ hypotenuse
tan θ = opposite ÷ adjacent
These ratios only apply to right-angled triangles, and θ must be one of the two acute angles.
这些比值只适用于直角三角形，而且 θ 必须是两个锐角之一。
2. SOH CAH TOA | SOH CAH TOA 记忆法
The easiest way to remember the three ratios is the phrase SOH CAH TOA. SOH means Sine equals Opposite over Hypotenuse, CAH means Cosine equals Adjacent over Hypotenuse, and TOA means Tangent equals Opposite over Adjacent.
记住这三个比值最简单的方法是口诀 SOH CAH TOA。SOH 表示正弦等于对边比斜边，CAH 表示余弦等于邻边比斜边，TOA 表示正切等于对边比邻边。
Write SOH CAH TOA at the top of your working in any trigonometry question. It helps you choose the correct ratio quickly and reduces careless mistakes.
在任何三角学题目中，先把 SOH CAH TOA 写在解题步骤的最上方。它能帮助你快速选择正确的比值，减少粗心错误。
3. Finding a Missing Side | 求未知边长
To find a missing side, first label the sides relative to the given angle. Then choose the ratio that links the known side and the unknown side.
要求未知边长，首先根据已知角标注各边。然后选择能联系已知边和未知边的三角比。
For example, if you know an angle of 35° and the hypotenuse is 12 cm, and you need the opposite side x, use sine because it involves opposite and hypotenuse.
例如，已知一个角为 35°，斜边为 12 cm，需要求对边 x，就使用正弦，因为正弦涉及对边和斜边。
sin 35° = x ÷ 12
x = 12 × sin 35°
Use your calculator in degree mode to obtain the value of sin 35°, then multiply by 12. Always give your final answer to a sensible degree of accuracy, usually three significant figures unless stated otherwise.
将计算器设置为度数模式，求出 sin 35° 的值，再乘以 12。最终答案通常保留三位有效数字，除非题目另有要求。
4. Finding a Missing Angle | 求未知角
When you know two sides and need to find an acute angle, use the inverse trigonometric functions sin⁻¹, cos⁻¹ or tan⁻¹.
当已知两条边并需要求一个锐角时，使用反三角函数 sin⁻¹、cos⁻¹ 或 tan⁻¹。
For example, if the opposite side is 5 cm and the hypotenuse is 8 cm, then sin θ = 5 ÷ 8. Use the inverse sine function to find θ.
例如，如果对边为 5 cm，斜边为 8 cm，那么 sin θ = 5 ÷ 8。使用反正弦函数来求 θ。
θ = sin⁻¹(5 ÷ 8) ≈ 38.7°
Before calculating, check that your calculator is in degree mode. If it is in radian mode, your answer will be wrong.
计算前，检查计算器是否处于度数模式。如果处于弧度模式，答案就会出错。
5. Exact Values for Special Angles | 特殊角的精确值
IGCSE students are expected to know the exact trigonometric values for 30°, 45° and 60° without using a calculator. These appear in non-calculator papers and in exact-value simplification questions.
IGCSE 学生需要在不使用计算器的情况下，掌握 30°、45° 和 60° 的精确三角值。这些值常出现在非计算器试卷和精确值化简题中。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
A common memory trick is to write sin 0°, 30°, 45°, 60°, 90° as √0/2, √1/2, √2/2, √3/2, √4/2, and reverse the order for cosine.
一个常见的记忆技巧是：把 sin 0°、30°、45°、60°、90° 写成 √0/2、√1/2、√2/2、√3/2、√4/2，余弦则顺序相反。
6. Angles of Elevation and Depression | 仰角与俯角
An angle of elevation is measured upwards from the horizontal, while an angle of depression is measured downwards from the horizontal. Both angles are always acute and are formed outside the object being observed.
仰角是从水平线向上测量的角，俯角是从水平线向下测量的角。这两个角始终是锐角，并且形成于被观察物体的外部。
When a person at point A looks up at an object at point B, the angle of elevation at A equals the angle of depression at B if both are measured from the same horizontal line and the line AB is straight.
当 A 点的观察者仰视 B 点的物体时，如果两者都从同一条水平线测量且 AB 为直线，则 A 处的仰角等于 B 处的俯角。
Draw a horizontal line at the observer’s eye level, then mark the angle and the relevant sides. This converts the description into a right-angled triangle you can solve.
在观察者视线高度画一条水平线，然后标出角和相关的边。这样就能把文字描述转化为可解的直角三角形。
7. Bearings and Trigonometry | 方位角与三角学
A bearing is an angle measured clockwise from the north direction. It is always written as three digits, such as 060° or 145°.
方位角是从正北方向顺时针测量的角。它总是写成三位数，例如 060° 或 145°。
Bearings questions often ask you to find a distance or a return bearing after a journey with two or more legs. Draw a clear north line at each point and construct right-angled triangles where possible.
方位角题目通常要求在经过两段或多段航程后，求某段距离或返回方位角。在每个点画清晰的北向线，并尽可能构造直角三角形。
For example, a boat travels 10 km on a bearing of 060°. Its eastward component is 10 sin 60° and its northward component is 10 cos 60°.
例如，一艘船沿方位角 060° 航行 10 km。它的东向分量为 10 sin 60°，北向分量为 10 cos 60°。
East = 10 sin 60° = 10 × √3/2 = 5√3 km
North = 10 cos 60° = 10 × 1/2 = 5 km
8. Using Pythagoras with Trigonometry | 毕达哥拉斯定理与三角学结合
Some questions require both Pythagoras’ theorem and trigonometric ratios. First use Pythagoras to find a missing side if two sides are known, then use trigonometry to find an angle, or vice versa.
有些题目需要同时使用毕达哥拉斯定理和三角比。如果已知两条边，先用毕达哥拉斯定理求出第三条边，再用三角比求角，或者反过来。
Remember that Pythagoras’ theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
记住，毕达哥拉斯定理指出：在直角三角形中，斜边的平方等于另外两条边的平方和。
c² = a² + b²
For example, if a triangle has sides 3 cm and 4 cm, the hypotenuse is 5 cm. You can then find an angle, such as θ = tan⁻¹(3 ÷ 4) ≈ 36.9°.
例如，一个三角形的两条直角边分别为 3 cm 和 4 cm，则斜边为 5 cm。然后你可以求角，例如 θ = tan⁻¹(3 ÷ 4) ≈ 36.9°。
9. Word Problems and Multi-Step Questions | 文字题与多步题
IGCSE trigonometry word problems often involve ladders leaning against walls, kites with strings, shadows cast by buildings, or planes taking off. The key skill is translating the words into a labelled diagram.
IGCSE 三角学文字题通常涉及靠在墙上的梯子、带线的风筝、建筑物投下的影子或起飞的飞机。关键技能是把文字转化为带标注的图形。
- Read the question twice and highlight the angle and the known length. | 仔细读题两遍，标出角度和已知边长。
- Draw a right-angled triangle and insert all given information. | 画出直角三角形，并填入所有已知信息。
- Label opposite, adjacent and hypotenuse relative to the angle. | 根据角度标出对边、邻边和斜边。
- Choose the correct ratio and write the equation. | 选择正确的三角比并列出方程。
- Solve, then check that the answer makes sense in the context. | 求解，然后检查答案在题目情境中是否合理。
Multi-step questions may require you to find one side first, then use that side to find another angle or length. Keep intermediate values in your calculator to avoid rounding errors.
多步题可能需要你先求出某条边，再用这条边求另一个角或另一条边。在计算器里保留中间值，避免舍入误差。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most common mistakes is using the wrong side as adjacent or opposite. Always label the sides from the angle you are using, not from the right angle.
最常见的错误之一是把邻边或对边标错。始终根据你正在使用的角来标注边，而不是从直角出发。
Another common error is using radian mode instead of degree mode. Before every trigonometry calculation, check the mode indicator on your calculator.
另一个常见错误是使用弧度模式而不是度数模式。每次进行三角计算前，都要检查计算器上的模式指示。
When an answer is required as an exact value, do not give a decimal approximation. For example, write √3 rather than 1.732.
当题目要求精确值时，不要给出小数近似值。例如，写 √3 而不是 1.732。
Finally, always include units in your final answer if the question uses units. A length should have cm, m or km, and an angle should have the degree symbol °.
最后，如果题目中使用单位，最终答案一定要包含单位。长度要写 cm、m 或 km，角度要写度数符号 °。
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