IB Mathematics: Trigonometry in Non-Right-Angled Triangles | IB数学：非直角三角形中的三角学

📚 IB Mathematics: Trigonometry in Non-Right-Angled Triangles | IB数学：非直角三角形中的三角学

Most students first meet trigonometry inside a right-angled triangle where sin, cos and tan are simple ratios of two sides. However, real-world problems – from finding the height of a tower to navigating a ship – rarely hand you a right angle. In IB Mathematics, both at Analysis and Approaches (AA) and Applications and Interpretation (AI), you must extend these tools to any triangle.

大多数同学最初接触三角学都是在直角三角形中，sin、cos、tan 只是两条边的简单比值。然而，现实世界中的问题——从测量塔的高度到船舶导航——很少会恰好给你一个直角。在 IB 数学中，无论是分析与方法（AA）还是应用与解释（AI），你都必须将这些工具推广到任意三角形。

This article gives a complete, exam-focused guide to the sine rule, cosine rule and area formula for non-right-angled triangles. We will show when each rule applies, how to avoid the famous “ambiguous case”, and how to answer IB-style questions with confidence.

本文将为你提供一份完整且紧扣考点的指南，覆盖正弦定理、余弦定理以及非直角三角形面积公式。我们会说明每个定理的适用条件、如何避开著名的“模糊情形”，以及如何自信地解答 IB 风格的问题。

1. Labelling a Triangle: Notation You Must Master | 三角形标记：必须掌握的记号

Before any rule can be used, you need to understand the standard notation used by the IB formula booklet and mark schemes. A triangle ABC is labelled so that vertex A is opposite side a, vertex B is opposite side b, and vertex C is opposite side c. In other words, side a is the side that does NOT touch vertex A.

在使用任何定理之前，你都需要理解 IB 公式手册和评分标准所采用的标准记号。在三角形 ABC 中，顶点 A 对着边 a，顶点 B 对着边 b，顶点 C 对着边 c。换句话说，边 a 是不经过顶点 A 的那条边。

This one-to-one correspondence between angles and opposite sides is the key that unlocks both the sine rule and the cosine rule. Whenever you write a ratio such as a/sin A, the angle A and the side a must lie opposite each other.

角与其对边之间的一一对应关系，是理解正弦定理与余弦定理的关键。每当你写出诸如 a/sin A 的比例式时，角 A 与边 a 必须互为对边。

Convention: side a opposite angle A, side b opposite angle B, side c opposite angle C

约定：边 a 对应角 A，边 b 对应角 B，边 c 对应角 C

Angles are usually written in degrees in non-calculator papers, but you must also be comfortable with radians for IB Analysis. ALWAYS check whether your calculator is in degree or radian mode before applying the rules – this single mistake costs thousands of IB marks every year.

在不使用计算器的试卷中，角度通常以度为单位，但在 IB 分析与方法课程中你也要熟悉弧度制。在应用定理前，务必检查计算器处于角度模式还是弧度模式——这个单一错误每年都让无数考生丢掉大量分数。

2. The Sine Rule | 正弦定理

The sine rule connects an angle with the side opposite to it. It has two equivalent forms: one for finding a side, and one for finding an angle. You should be comfortable switching between them.

正弦定理将一个角与其对边联系起来。它有两条等价的形式：一种用于求边，一种用于求角。你应当能够熟练地在二者之间切换。

Sine rule (finding a side): a/sin A = b/sin B = c/sin C

正弦定理（求边）：a/sin A = b/sin B = c/sin C

Use this form when you know one full “angle-side pair” and one other side OR one other angle. For example, if you know angle A and side a, plus angle B, you can find side b.

当你已知一组完整的“角对边”以及另一条边或另一个角时，使用这个形式。例如，已知角 A 与边 a，再知道角 B，就可以求出边 b。

Sine rule (finding an angle): sin A/a = sin B/b = sin C/c

正弦定理（求角）：sin A/a = sin B/b = sin C/c

Use the reciprocal form when you know two sides and one of the opposite angles. For instance, if sides a and b are known and angle B is known, you can solve for angle A.

当你已知两条边以及其中一边的对角时，使用倒数形式。例如，已知边 a、b 以及角 B，就可以解出角 A。

To apply the sine rule, you only need two of the three fractions. The rule works for ANY triangle, including right-angled and obtuse-angled triangles, which makes it far more powerful than simple SOHCAHTOA.

要应用正弦定理，只需要三个比例中的两个。它对任何三角形都成立，包括直角三角形和钝角三角形，因此它远比简单的 SOHCAHTOA 更强大。

3. The Cosine Rule: Finding a Side | 余弦定理：求边

The cosine rule is the generalised version of Pythagoras’ theorem. When the included angle is 90°, cos 90° = 0, and the rule collapses exactly into a² = b² + c². This is a useful memory check in exams.

余弦定理是勾股定理的推广。当夹角为 90° 时，cos 90° = 0，此时公式恰好退化为 a² = b² + c²。这在考试中是一个有用的检验方法。

Cosine rule (side): a² = b² + c² − 2bc cos A

余弦定理（求边）：a² = b² + c² − 2bc cos A

Here angle A is the INCLUDED angle between side b and side c. The rule is cyclic, so the same formula can be rotated: b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C.

其中角 A 是边 b 与边 c 之间的夹角。该公式具有轮换对称性：b² = a² + c² − 2ac cos B，c² = a² + b² − 2ab cos C。

Use the cosine rule for sides when you are given two sides and the included angle (this combination is abbreviated SAS, “side-angle-side”). For example, in triangle ABC, if b = 7 cm, c = 5 cm and angle A = 40°, you cannot use the sine rule because no full angle-side pair is known. The cosine rule is your only direct option.

当已知两边及其夹角（简称 SAS，即“边角边”）时，使用余弦定理求边。例如，在三角形 ABC 中，若 b = 7 cm，c = 5 cm，角 A = 40°，此时无法使用正弦定理，因为不存在完整的“角对边”组合。余弦定理是你唯一直接的选择。

Take the square root of both sides to finish. Remember that a length is positive, so you ignore negative roots. In exact-value questions, leave your answer in the form √k rather than using a decimal.

最后两边开平方根。记住长度为正，因此要舍去负根。在精确值题目中，把答案保留为 √k 的形式，而不要写成小数。

4. The Cosine Rule: Finding an Angle | 余弦定理：求角

By rearranging the side version of the cosine rule, we can solve for an angle when all three sides are known (SSS, “side-side-side”). This is a required skill in IB Mathematics AA and AI.

将余弦定理的求边形式重新整理，就可以在已知三边（SSS，即“边边边”）的情况下解出角度。这是 IB 数学 AA 和 AI 课程中的一项必备技能。

Cosine rule (angle): cos A = (b² + c² − a²) / (2bc)

余弦定理（求角）：cos A = (b² + c² − a²) / (2bc)

Notice that the numerator contains the square of the side opposite to the angle you want, subtracted from the sum of squares of the other two sides. For a triangle with sides a = 6, b = 7, c = 8, the largest angle is opposite the largest side c, so you would expect angle C to be the largest.

注意，分子中是所求角的对边平方，另两边平方和减去该平方。对于边 a = 6，b = 7，c = 8 的三角形，最大的角对着最大的边 c，因此可以预计角 C 最大。

One major advantage of the cosine rule over the sine rule is that cos A is unique between 0° and 180°. For any angle in this range, the cosine function is one-to-one, so the inverse cosine gives exactly one answer. This means the ambiguous case never occurs when you use the cosine rule to find an angle.

余弦定理相比正弦定理的一大优势在于：在 0° 到 180° 之间，cos A 的结果是唯一的。在该范围内，余弦函数是一一对应的，因此反余弦只会给出一个答案。这意味着用余弦定理解角永远不会出现“模糊情形”。

If cos A comes out negative, then angle A is obtuse (greater than 90°). If it comes out positive, angle A is acute. If it equals zero, the triangle contains a right angle. These signs give you a built-in sanity check.

如果 cos A 为负值，则角 A 为钝角（大于 90°）；如果为正值，则角 A 为锐角；如果等于零，则三角形中含有直角。这些符号为你提供了内在的合理性检验。

5. Area of a Non-Right-Angled Triangle | 非直角三角形面积

You already know area = ½ × base × height, but in a non-right-angled triangle the height is rarely given. A second area formula uses two sides and the INCLUDED angle, and it is printed in the IB formula booklet.

你已经知道面积 = ½ × 底 × 高，但在非直角三角形中，高度往往不会直接给出。第二个面积公式使用两边及其夹角，该公式印在 IB 公式手册中。

Area = ½ ab sin C = ½ ac sin B = ½ bc sin A

面积 = ½ ab sin C = ½ ac sin B = ½ bc sin A

In words: the area is half the product of two sides multiplied by the sine of the angle between them. This formula works because b sin C is exactly the perpendicular height drawn from vertex A to side b.

用语言来描述：面积等于两条边乘积的一半再乘以它们夹角的正弦。此公式之所以成立，是因为 b sin C 正是从顶点 A 向边 b 所作垂线的高度。

In contexts such as surveying and design (IB Applications), this formula lets you find the area of any polygon by splitting it into triangles. Two adjacent sides and one diagonal create a triangle; applying the area rule to each triangular piece and adding the results gives the total area.

在测量与设计等应用场景（IB 应用与解释）中，这个公式允许你将任意多边形分割成三角形来求面积。以两个相邻边和一条对角线构成一个三角形，对每一块三角形用面积公式再求和，即可得到总面积。

If you need an exact area, keep the sine term exact. For example, ½ × 5 × 8 × sin 30° = 20 × ½ = 10 exactly. Recognising special angles (30°, 45°, 60°) helps you solve non-calculator questions quickly.

如果需要精确面积，就要保留正弦项的精确值。例如，½ × 5 × 8 × sin 30° = 20 × ½ = 10 为精确值。熟悉特殊角（30°、45°、60°）的正弦值有助于快速解决无计算器题目。

6. Choosing the Correct Rule: A Decision Strategy | 定理选择策略

Examiners deliberately design problems so that only ONE rule is efficient. If you start with the wrong rule, you might still obtain the answer but with far more work and a higher chance of error. The table below summarises the decision process.

命题者会刻意设计题目，使通常只有一种定理是高效的。如果一开始选错定理，虽然仍可能得到答案，但计算量更大，出错率也更高。下表总结了决策流程。

Given information / 已知条件	Rule to use / 使用定理
Two angles + one side (AAS or ASA) / 两角一边	Sine rule / 正弦定理
Two sides + non-included angle (SSA) / 两边及其中一边对角	Sine rule, then check for ambiguity / 正弦定理，随后检查模糊情形
Two sides + included angle (SAS) / 两边及其夹角	Cosine rule for the third side / 余弦定理求第三边
Three sides (SSS) / 三边	Cosine rule for angles / 余弦定理求角
Area + two sides / 面积及两边	Area = ½ ab sin C / 面积公式

When you know two angles first, immediately find the third angle using the angle sum A + B + C = 180°. The sine rule then requires no calculator-based guesswork, and this often opens the door to the rest of the question.

当你知道两个角时，先用三角形内角和 A + B + C = 180° 求出第三个角。此时正弦定理无需猜测，往往也能为后续解答打开大门。

When you know two sides and the included angle, NEVER try to use the sine rule first. You do not yet know any opposite angle. The cosine rule is mandatory for finding that third side.

当你知道两边及其夹角时，绝不要先尝试正弦定理。因为此时你还不知道任何对角。求第三条边必须使用余弦定理。

7. The Ambiguous Case of the Sine Rule | 正弦定理的模糊情形

This is the single most examined subtlety of non-right triangle trigonometry in IB. When you solve for an angle using the sine rule and you know two sides plus a non-included angle (SSA), the unknown angle sine may correspond to TWO possible angles: one acute and one obtuse.

这是 IB 非直角三角形三角学中最受重视的微妙之处。当你用正弦定理解角，且已知两边及其中一边的对角（SSA）时，所求角的正弦值可能对应两个角度：一个是锐角，一个是钝角。

For example, if sin B = 0.5 from the sine rule, then B could be 30° or B could be 150°, because sin 30° = sin 150° = 0.5. Both angles lie between 0° and 180°. Which one is correct depends on the geometry of the triangle.

例如，若通过正弦定理得到 sin B = 0.5，则 B 可能是 30°，也可能是 150°，因为 sin 30° = sin 150° = 0.5。两个角都在 0° 到 180° 之间。哪一个正确取决于三角形的几何条件。

The IB mark scheme expects you to show the ambiguous case explicitly. If you simply press sin⁻¹ on your calculator, you only recover the acute angle. You must subtract it from 180° to obtain the obtuse candidate, then test whether that candidate still leaves a positive third angle.

IB 评分标准期望你明确展示模糊情形。如果只按计算器上的 sin⁻¹，你只会得到锐角。你必须用 180° 减去它得到钝角候选值，然后检验该候选值是否仍能让第三个角为正。

If A + B_obtuse < 180°, the obtuse triangle is valid: two possible triangles exist.

若 A + 钝角B < 180°，则钝角三角形成立：存在两种可能的三角形。
If A + B_obtuse ≥ 180°, the obtuse case is impossible: only the acute solution is kept.

若 A + 钝角B ≥ 180°，则钝角情形不可能：只保留锐角解。

The source of the ambiguity is that a given chord of a circle subtends equal angles at points on the same arc. The sine function is symmetric about 90°, so sin θ = sin(180° − θ) for all θ. Always write down both candidates before choosing.

模糊性的根源在于：固定弦在圆弧上同侧各点所张的角相等。正弦函数关于 90° 对称，因此对所有 θ 都有 sin θ = sin(180° − θ)。在作出选择前务必写出两个候选值。

A wise habit: once you think you have found an angle, check that the largest side sits opposite the largest angle. If your solution violates this monotonic relationship, you have chosen the wrong candidate.

一个明智的习惯：当你认为已求出某个角时，检查“大边对大角”是否成立。如果你的解答违反了这一单调关系，说明你选错了候选值。

8. Worked Example: Solving a Mixed Problem | 综合例题精讲

Consider a triangle ABC with b = 12 cm, c = 8 cm and angle B = 42°. Find side a, angle C, angle A, and the area of the triangle.

考虑三角形 ABC，其中 b = 12 cm，c = 8 cm，角 B = 42°。求边 a、角 C、角 A 以及三角形面积。

Step 1 – Find angle C using the sine rule.

第一步——用正弦定理求角 C。

sin C/c = sin B/b → sin C = c sin B/b = 8 sin 42°/12 ≈ 0.4462

Calculator gives C ≈ 26.5°. The obtuse candidate is 180° − 26.5° = 153.5°. Adding B = 42° gives 42° + 153.5° = 195.5° > 180°, so the obtuse candidate is rejected. Thus C ≈ 26.5°.

计算器给出 C ≈ 26.5°。钝角候选值为 180° − 26.5° = 153.5°。将 B = 42° 代入，42° + 153.5° = 195.5° > 180°，因此钝角候选值被舍去。故 C ≈ 26.5°。

Step 2 – Find angle A by angle sum.

第二步——用内角和求角 A。

A = 180° − 42° − 26.5° ≈ 111.5°

Angle A is obtuse, which immediately tells us the triangle is obtuse-angled. Notice that side a must now be the largest side.

角 A 为钝角，这立即说明三角形为钝角三角形。注意此时边 a 必为最大边。

Step 3 – Find side a using the sine rule again.

第三步——再次使用正弦定理求边 a。

a/sin A = b/sin B → a = b sin A/sin B = 12 sin 111.5°/sin 42° ≈ 16.7 cm

Step 4 – Find the area. The two known sides that touch angle A or angle B can be used. Use sides b and c with included angle A, or sides a and b with included angle C. Since a and c are adjacent to angle B, area = ½ × a × c × sin B.

第四步——求面积。可以使用与角 A 或角 B 相邻的已知两边。使用边 b、c 与夹角 A，或使用边 a、c 与夹角 B。因为 a 与 c 相邻于角 B，面积 = ½ × a × c × sin B。

Area = ½ × 16.7 × 8 × sin 42° ≈ 44.7 cm²

面积 = ½ × 16.7 × 8 × sin 42° ≈ 44.7 cm²

In an IB exam, always write out the formula you are applying before substituting numbers. The “method” mark is awarded for the formula even if your substitution contains a small arithmetic slip.

在 IB 考试中，务必在代入数字前先写出所应用的公式。即使你的代入中存在轻微运算错误，展示公式仍然能获得“方法分”。

9. Bearings, Elevation and Real-World Contexts | 方位角、仰角与现实情境

IB Applications and Interpretation papers frequently wrap triangle trigonometry in real-world language. A “bearing” is an angle measured clockwise from north, usually written as a three-digit number such as 047°. Two bearings and a distance can define a triangle that requires the cosine or sine rule to solve.

IB 应用与解释试卷经常把三角形三角学包装在现实情境中。“方位角”是从正北方向顺时针测量的角度，通常写成三位数，例如 047°。两个方位角和一段距离可以定义一个需要用正弦或余弦定理求解的三角形。

For example, a ship sails 40 km on a bearing of 060°, then changes course and sails 60 km on a bearing of 150°. The change in direction at the turning point is the difference of the two bearings: 150° − 60° = 90°. Here the included angle is exactly 90°, so Pythagoras works as a special case.

例如，一艘船沿方位角 060° 航行 40 km，然后改变航向，沿方位角 150° 继续航行 60 km。在转向点，方向的改变量是两个方位角之差：150° − 60° = 90°。此时夹角恰好为 90°，勾股定理可作为特例使用。

If the second bearing were 135° instead, the included angle would be 135° − 60° = 75°, and the final displacement from the start would require the cosine rule: d² = 40² + 60² − 2 × 40 × 60 × cos 75°.

如果第二个方位角改为 135°，则夹角为 135° − 60° = 75°，此时起点到终点的最终位移就需要用余弦定理：d² = 40² + 60² − 2 × 40 × 60 × cos 75°。

Similarly, angles of elevation and depression create right-angled triangles only if the observer and the object are horizontally aligned. In three-dimensional problems from IB AAHL, you may need to project a solid onto a plane to create a solvable non-right triangle; the same sine and cosine rules then apply.

类似地，只有当观测者与物体处于同一水平线时，仰角与俯角才会形成直角三角形。在 IB AAHL 的三维问题中，你可能需要将立体投影到平面上，以构造一个可解的非直角三角形；此时同样应用正弦定理和余弦定理。

When solving word problems, always draw a clear diagram first. Mark every known side and angle, and identify the triangle you will solve. In IB AI, one mark is often reserved for the correct interpretation of the context, not just for the formula work.

在解答应用题时，一定要先画清晰示意图。标出所有已知边与角，并确定你要解的三角形。在 IB AI 考试中，经常有一分专门对应情境的正确理解，而不仅仅是公式运算。

10. Common Errors and Exam Tactics | 常见错误与考试策略

Error 1: Using the sine rule with the wrong pair. The sine rule only works when the side and angle in each ratio are opposite each other. If you match side a with angle B, every result will be nonsense.

错误一：正弦定理中角与边配对错误。正弦定理只对“边与对角成比例”才成立。如果你把边 a 与角 B 配对，所有结果都会出错。

Error 2: Forgetting the ambiguous case. Every SSA sine-rule question in IB has a hidden test: candidates must justify why only one solution remains. Show the acute answer, generate the obtuse answer, and state the rejection reason.

错误二：忽略模糊情形。IB 中每道 SSA 型正弦定理题目都暗含一个测试：考生必须说明为何仅剩一个解。写出锐角答案，再生成钝角答案，并陈述舍去理由。

Error 3: Incorrect calculator mode. If a question expects degrees and your calculator is in radians, sin 42° will be computed incorrectly. Train yourself to check the DEG or RAD indicator before every computation.

错误三：计算器模式错误。如果题目要求角度制而计算器处于弧度制，sin 42° 就会出错。请养成在每次计算前检查 DEG 或 RAD 指示符的习惯。

Error 4: Rounding too early. In multi-step problems, always keep full calculator precision in intermediate values. Round only the final answer to the required number of significant figures, typically 3 s.f. in IB questions.

错误四：过早四舍五入。在多步计算中，中间值应保留计算器的完整精度。只对最终答案按题目要求四舍五入，IB 通常要求保留 3 位有效数字。

Error 5: Misidentifying the included angle in the area formula. The angle in Area = ½ ab sin C must physically sit between sides a and b. If you use an opposite angle, the area will be wrong. Quickly verify by sketching which angle the two sides share.

错误五：面积公式中夹角识别错误。Area = ½ ab sin C 中的角 C 必须物理上位于边 a 与边 b 之间。如果误用对角，面积就会出错。快速画图确认两条边所共用的角即可验证。

Finally, use estimation as a check. If side a is opposite a 110° angle, it should be the longest side. If your computed side is shorter than another side whose opposite angle is 30°, you have almost certainly made a mistake.

最后，用估算作为检验手段。如果边 a 对着 110° 的角，它应当是最长边。如果你算出的边反而比某个对应角仅为 30° 的边还短，那你几乎肯定出了错。

11. Connecting to the IB Formula Booklet | 与 IB 公式手册的衔接

All of the formulas in this article are provided in the IB formula booklet, but remembering when and how to use each one is entirely up to you. The formula booklet gives three equivalent sine-rule fractions and two versions of the cosine rule; it does not tell you which triangle to solve first.

本文中所有公式都可以在 IB 公式手册中找到，但何时使用、如何使用完全取决于你自己。公式手册给出了正弦定理的三个等价比例式以及余弦定理的两种形式，但它不会告诉你应该先解哪个三角形。

On the exam, underline the given quantities in the question and match them to the four scenarios: AAS/ASA, SAS, SSA, SSS. This small act of categorisation immediately tells you which page of your mental formula booklet to open.

在考试中，划出题目给出的量，并将它们归入四种情形：AAS/ASA、SAS、SSA、SSS。这个小小的分类动作能立刻告诉你应该调用脑中公式手册的哪一页。

For non-calculator papers, know the common values: sin 30° = cos 60° = ½, sin 45° = cos 45° = √2/2, sin 60° = cos 30° = √3/2, and sin 90° = 1. These appear repeatedly in exact-value questions built around ½ ab sin C and the cosine rule.

对于无计算器试卷，要熟记常见值：sin 30° = cos 60° = ½，sin 45° = cos 45° = √2/2，sin 60° = cos 30

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IB Mathematics: Trigonometry in Non-Right-Angled Triangles | IB数学：非直角三角形中的三角学