📚 IGCSE Maths: Sine Rule, Cosine Rule and Area of a Triangle | IGCSE 数学：正弦定理、余弦定理与三角形面积
When you move beyond right-angled trigonometry, many IGCSE questions ask you to find missing sides or angles in any triangle. The sine rule, cosine rule and the area formula give you a complete toolkit for non-right-angled triangles, and they appear again and again in both Core and Extended papers, in bearing problems, in surveying questions and inside 3D solids.
当你从直角三角形三角学进入更广泛的内容时，许多 IGCSE 题目要求你求任意三角形中的未知边或角。正弦定理、余弦定理和面积公式为你提供了处理非直角三角形的完整工具，它们会反复出现在核心卷与扩展卷中，出现在方位角问题、测量问题以及三维立体图形内部。
1. Why we need the sine and cosine rules | 为什么需要正弦定理和余弦定理
In a right-angled triangle, Pythagoras' theorem and the basic ratios sin, cos and tan are enough. However, most triangles in real life and in IGCSE exam questions do not contain a right angle at all.
在直角三角形中，勾股定理和基本的 sin、cos、tan 比已经足够。然而，现实生活和 IGCSE 考题中的大多数三角形根本不含直角。
The sine rule and the cosine rule work for any triangle, so they are essential for problems involving non-right-angled shapes such as bearings, navigation, surveying, forces and 3D geometry.
正弦定理和余弦定理适用于任何三角形，因此对于方位角、航海、测量、力的合成以及三维几何等涉及非直角图形的问题，它们是必不可少的。
Questions on this topic often appear in both the core and extended IGCSE papers, especially when a triangle is given with side lengths and angles in non-standard positions, or when a diagram must first be extracted from a written description of a journey or a field.
这一主题的题目经常出现在 IGCSE 核心卷和扩展卷中，尤其是当三角形以非标准位置给出边长和角度时，或者当图形需要先从一段关于行程或田地的文字描述中提取出来时。
Another reason these rules matter is that they let you solve a triangle completely: once you know three suitable pieces of information, you can find every remaining side and angle, and then work out the area.
这些定理重要的另一个原因是，它们让你能够完整地解出三角形：一旦你已知三个合适的信息，你就能求出其余所有的边和角，进而算出面积。
2. Labelling triangles and standard notation | 三角形标记与标准符号
In IGCSE trigonometry, vertices are usually labelled A, B and C. The side opposite vertex A is called a, the side opposite B is b, and the side opposite C is c.
在 IGCSE 三角学中，顶点通常标记为 A、B 和 C。顶点 A 的对边称为 a，顶点 B 的对边称为 b，顶点 C 的对边称为 c。
Angles are written as capital letters and sides as lower-case letters. This convention makes the sine rule and the cosine rule much easier to remember and apply, because every side is automatically paired with its opposite angle.
角用大写字母表示，边用小写字母表示。这一约定使正弦定理和余弦定理更容易记忆和应用，因为每条边都自动与其对角配对。
Always draw a clear diagram and place each side opposite its matching angle. A correct diagram is one of the simplest ways to avoid careless mistakes later on, especially when an angle is obtuse and the sketch tempts you to place it incorrectly.
一定要画出清晰的图形，并把每条边放在其对应角的对边位置。正确的图形是避免后续粗心错误最简单的方法之一，尤其是当某个角是钝角、而草图容易诱导你把它画错位置时。
des always satisfy the triangle inequalities: each side must be shorter than the sum of the other two. Checking these two facts first will quickly reveal impossible or contradictory data in an exam question.
三角形的三个角始终满足 A + B + C = 180°，三条边始终满足三角不等式：任意一边必须小于另外两边之和。先检查这两点，能迅速发现考题中不可能成立或自相矛盾的数据。
3. The sine rule | 正弦定理
For any triangle ABC, the ratio of a side to the sine of its opposite angle is constant. This is the sine rule:
对于任意三角形 ABC，每条边与其对角正弦的比值是常数，这就是正弦定理：
a ÷ sin A = b ÷ sin B = c ÷ sin C
The rule can also be written upside down, which is the form you use when you are looking for an angle rather than a side:
该定理也可以倒过来写，当你要求角而不是求边时，用的就是这种形式：
sin A ÷ a = sin B ÷ b = sin C ÷ c
Where does it come from? A sin B gives a ÷ sin A = b ÷ sin B, and the third ratio follows in exactly the same way.
它从哪里来？从顶点 C 向边 c 作垂线，设其长度为 h。于是 h = b sin A，同时 h = a sin B，所以 b sin A = a sin B。两边同除以 sin A sin B，就得到 a ÷ sin A = b ÷ sin B，第三个比值也可用完全相同的方法得到。
Notice the key requirement: the sine rule always needs a complete matching pair, that is, one side together with its opposite angle. Without a matching pair, you cannot start the sine rule at all.
请注意关键条件：使用正弦定理必须有一条完整的"对应边与对角"配对信息。没有这样的配对，你根本无法开始用正弦定理。
4. Using the sine rule to find a side | 用正弦定理求边
To find a missing side, write down the two ratios that involve the information you have and the side you want, then rearrange and substitute.
要求一条未知边，先写出包含已知信息和所求边的两个比值，然后变形并代入。
ABC, A = 35°, B = 80° and a = 12 cm. Find b, giving your answer to 3 significant figures.
例题：在三角形 ABC 中，A = 35°，B = 80°，a = 12 cm。求 b，答案保留 3 位有效数字。
b ÷ sin 80° = 12 ÷ sin 35°
So b = 12 × sin 80° ÷ sin 35° = 12 × 0.9848 ÷ 0.5736 ≈ 20.6 cm.
因此 b = 12 × sin 80° ÷ sin 35° = 12 × 0.9848 ÷ 0.5736 ≈ 20.6 cm。
Notice how the multiplication sits on the top line: when the unknown side is on the top of its fraction, you multiply by the sine of the angle opposite the unknown side and divide by the sine of the angle opposite the known side.
注意乘法在分子上：当未知边位于其分数的分子位置时，你乘以"未知边的对角"的正弦，再除以"已知边的对角"的正弦。
le. This one extra line is extremely common in mark schemes and earns an easy method mark.
有时第三个角没有直接给出。此时先用 A + B + C = 180° 求出它，再使用正弦定理。这一步额外计算在评分标准中极为常见，可以轻松拿到方法分。
5. Using the sine rule to find an angle (and the ambiguous case) | 用正弦定理求角与"两解"问题
a = sin B ÷ b so that the unknown sine stays on the top line, then apply sin⁻¹ at the end.
当未知量是角时，使用倒置形式 sin A ÷ a = sin B ÷ b，使未知的正弦留在分子位置，最后再取 sin⁻¹。
Worked example: a = 8 cm, b = 11 cm and A = 40°. Find angle B.
例题：a = 8 cm，b = 11 cm，A = 40°。求角 B。
sin B ÷ 11 = sin 40° ÷ 8
So sin B = 11 × sin 40° ÷ 8 = 11 × 0.6428 ÷ 8 = 0.8839, giving B ≈ 62.1°.
因此 sin B = 11 × sin 40° ÷ 8 = 11 × 0.6428 ÷ 8 = 0.8839，得到 B ≈ 62.1°。
117.9°. Both values are mathematically possible when only the data above is given, because both leave room for the remaining angle C to be positive.
这里正是考官最爱设置的陷阱：sin B = sin (180° − B)，所以 B ≈ 62.1° 或 B ≈ 117.9°。仅凭上述数据，这两个值在数学上都可能成立，因为两者都让剩余角 C 仍为正角。
, so this triangle is also valid. If a second possibility gives a total of 180° or more, you must reject it.
如何判断？逐一检验：62.1° + 40° = 102.1° < 180°，该三角形成立；117.9° + 40° = 157.9° < 180°，该三角形同样成立。如果第二种可能使角度和达到或超过 180°，就必须舍弃它。
In most IGCSE questions the diagram tells you whether the angle is acute or obtuse, so always look at the shape before writing your final answer. If the diagram shows B as an obtuse angle, choose 117.9°; if it shows B as acute, choose 62.1°.
在大多数 IGCSE 题目中，图形会告诉你该角是锐角还是钝角，所以在写最终答案前一定要先看图。如果图中 B 是钝角，就选 117.9°；如果图中 B 是锐角，就选 62.1°。
6. The cosine rule | 余弦定理
The cosine rule handles the two situations the sine rule cannot: three sides known with no angle, or two sides known with the angle between them (the included angle).
余弦定理处理正弦定理无法解决的两种情况：已知三边而无任何角，或者已知两边及其夹角。
a² = b² + c² − 2bc cos A
To find an angle, rearrange the same relationship so that cos A is the subject:
要求一个角，就把同一关系变形，使 cos A 成为主项：
cos A = (b² + c² − a²) ÷ (2bc)
Two details decide whether you get the marks. First, the angle A must be the angle between sides b and c, not any other angle. Second, the side on the left must be the side opposite that included angle.
有两个细节决定你能否拿分。第一，角 A 必须是边 b 与边 c 之间的夹角，而不是其他角。第二，等号左边的边必须是该夹角所对的边。
n A = 90°, cos 90° = 0 and the rule collapses to a² = b² + c², which is exactly Pythagoras' theorem. So the cosine rule is a generalisation of Pythagoras, and questions that say 'show that angle ABC is not a right angle' are really asking you to compare a² with b² + c².
一个有用的检验：当 A = 90° 时，cos 90° = 0，定理退化为 a² = b² + c²，这正是勾股定理。所以余弦定理是勾股定理的推广；而题目中"证明角 ABC 不是直角"这类要求，实际上就是让你比较 a² 与 b² + c² 的大小。
7. Using the cosine rule to find a side | 用余弦定理求边
Worked example: b = 7 cm, c = 9 cm and the included angle A = 60°. Find a.
例题：b = 7 cm，c = 9 cm，夹角 A = 60°。求 a。
a² = 7² + 9² − 2 × 7 × 9 × cos 60°
a² = 49 + 81 − 126 × 0.5 = 130 − 63 = 67, so a = √67 ≈ 8.19 cm.
a² = 49 + 81 − 126 × 0.5 = 130 − 63 = 67，所以 a = √67 ≈ 8.19 cm。
Watch the signs carefully. If the included angle is obtuse, cos A is negative, so the term −2bc cos A becomes positive and the answer side is longer than either of the other two sides. That is a good sanity check.
要特别小心符号。如果夹角是钝角，cos A 为负，于是 −2bc cos A 这一项变成正的，所求边就会比另外两边都长。这是一个很好的合理性检验。
+ 60 = 196, so a = 14 cm exactly, a classic exam answer that shows the method is correct.
再看一个有用的例子：两边长 6 cm 与 10 cm，夹角 120°。则 a² = 36 + 100 − 120 × (−0.5) = 136 + 60 = 196，所以 a = 14 cm，恰好是整数，这是能验证方法正确的经典考题答案。
Take the square root at the very end, never in the middle, and keep the full calculator value on your screen until the final rounding step. Rounding to 3 significant figures too early is a very common cause of lost accuracy marks.
平方根要在最后一步才取，绝不在中间步骤取；并在屏幕上保留完整的计算器数值，直到最后一步才四舍五入。过早保留 3 位有效数字，是失去精度分最常见的原因。
8. Using the cosine rule to find an angle | 用余弦定理求角
When all three sides are known and no angle is given, you must use the rearranged cosine rule. This situation cannot be solved with the sine rule.
当已知三边而没有任何角时，必须使用变形后的余弦定理。这种情况无法用正弦定理解决。
Worked example: a = 5 cm, b = 6 cm, c = 7 cm. Find angle A.
例题：a = 5 cm，b = 6 cm，c = 7 cm。求角 A。
cos A = (6² + 7² − 5²) ÷ (2 × 6 × 7) = (36 + 49 − 25) ÷ 84 = 60 ÷ 84 = 0.7143
7143) ≈ 44.4°, which is consistent with the largest angle being opposite the longest side.
所以 A = cos⁻¹(0.7143) ≈ 44.4°，这与"最长边对角最大"的规律一致。
Always put the side opposite the angle you want on the top line of the numerator, and the two sides forming that angle in the denominator. A quick way to remember the numerator is 'the two sides next to the angle, squared and added, minus the opposite side squared'.
一定要把"所求角的对边"放在分子式被减数位置，把"组成该角的两边"放在分母。记分子的快速方法是："夹角两边平方相加，减去对边平方"。
A good accuracy habit is to check that cos A lies between −1 and 1. If your calculator shows a maths error, you have almost certainly substituted the sides into the wrong positions or mis-ordered the subtraction.
良好的精度习惯是检查 cos A 是否介于 −1 与 1 之间。如果计算器显示 maths error，几乎可以肯定你代错了位置或把减法顺序弄反了。
9. Area of a triangle = ½ab sin C | 三角形面积公式 ½ab sin C
The IGCSE area formula for a non-right-angled triangle is:
IGCSE 中非直角三角形的面积公式是：
Area = ½ × a × b × sin C
Here a and b are any two sides of the triangle and C is the angle between them. The same idea can be written in three equivalent ways:
其中的 a 与 b 是三角形的任意两边，C 是它们之间的夹角。同一思想可写成三种等价形式：
Area = ½ab sin C = ½bc sin A = ½ac sin B
Why does it work? rpendicular from the opposite vertex). Substituting gives Area = ½b × a sin C = ½ab sin C.
它为什么成立？从熟悉的公式"面积 = ½ × 底 × 垂直高"出发。取 b 为底，垂高为 h = a sin C（从对角顶点作垂线可得）。代入得 面积 = ½b × a sin C = ½ab sin C。
and the included angle C = 35°. Then Area = ½ × 8 × 11 × sin 35° = 44 × 0.5736 ≈ 25.2 cm².
例题：a = 8 cm，b = 11 cm，夹角 C = 35°。则 面积 = ½ × 8 × 11 × sin 35° = 44 × 0.5736 ≈ 25.2 cm²。
t-angled triangle the formula reduces to ½ab, the ordinary half-base-times-height result. Also note that sin of an obtuse angle is still positive, so the area always comes out positive even when the included angle is greater than 90°.
记住 sin 90° = 1，所以对直角三角形，该公式退化为 ½ab，即普通的"底乘高的一半"。同时注意钝角的正弦仍然为正，因此即使夹角大于 90°，面积也始终为正。
The area formula is also the starting point for the proof of the sine rule, which is a favourite 'show that' question in the Extended paper.
面积公式同时也是证明正弦定理的起点，这也是扩展卷中最受偏爱的"证明题"之一。
10. Choosing the right rule | 如何选择正确的定理
Many marks are lost not by bad algebra but by choosing the wrong rule. Use this table as a decision guide before you write anything down.
许多失分并不是因为代数运算出错，而是因为选错了定理。在动笔之前，先用下表作为判断指南。
| What you are given | 已知条件 | What to use | 应使用 |
| Two angles and any side | 两角及任意一边 | Sine rule (find a side) | 正弦定理（求边） |
| Two sides and an angle opposite one of them | 两边及其中一边的对角 | Sine rule (find an angle, check the ambiguous case) | 正弦定理（求角，须检验两解） |
| Two sides and the included angle | 两边及其夹角 | Cosine rule (find the third side) | 余弦定理（求第三边） |
| Three sides, no angle | 三边，无角 | Cosine rule rearranged (find an angle) | 变形余弦定理（求角） |
| Two sides and the included angle, asked for area | 两边及夹角，求面积 | Area = ½ab sin C | 面积 = ½ab sin C |
| A right angle is present | 存在直角 | SOHCAHTOA or Pythagoras (usually faster) | 用 SOHCAHTOA 或勾股定理（通常更快） |
A second habit that saves time: count what you know. If a triangle has three known pieces of data, including at least one side, it can normally be solved completely. If you know only three angles, the triangle can be scaled infinitely and no side length can be determined.
另一个能节省时间的习惯是"数已知量"。如果一个三角形有三个已知条件，且其中至少有一条边，通常就能完整解出。如果只知道三个角，则三角形可以无限缩放，任何边长都无法确定。
Finally, remember that a single problem may need more than one rule. A common pattern is: use the cosine rule in one triangle to find a shared side, then use the sine rule in the neighbouring triangle to find an angle, and finish with the area formula.
最后，记住一道题可能需要不止一个定理。常见的模式是：先在一个三角形中用余弦定理求出公共边，再在相邻三角形中用正弦定理求角，最后用面积公式收尾。
11. Bearings, navigation and 3D problems | 方位角、航行与三维问题
Bearings are measured clockwise from north and written with three digits, for example 047° or 210°. In bearing questions the sine and cosine rules are usually applied to the triangle formed by the starting point, the turning point and the destination.
方位角从正北方向顺时针量度，写成三位数，例如 047° 或 210°。在方位角问题中，正弦定理和余弦定理通常用于由起点、转向点和终点构成的三角形。
Two geometry facts do most of the work. First, alternate angles between two north lines are equal, which lets you transfer a bearing into an interior angle of the triangle. Second, the interior angle at a point between two given bearings is found by subtracting the smaller bearing from the larger, or by using 360° minus that difference.
有两条几何事实承担了大部分工作。第一，两条正北方向线之间的内错角相等，由此可以把方位角转换成三角形的内角。第二，某个顶点处两条已知方位线之间的内角，可以用较大的方位角减去较小的，或者用 360° 减去该差值得到。
In 3D questions, the technique is to break the solid into flat triangles. A typical approach: find the length of a space diagonal inside a base triangle using the cosine rule, then use the sine rule or the area formula in a vertical triangle containing that line.
在三维问题中，技巧是把立体图形拆成平面三角形。典型做法是：先用余弦定理求出底面三角形内部某条对角线的长度，再在包含该线的竖直三角形中使用正弦定理或面积公式。
For shortest distance from a point to a line, or the angle between a line and a plane, always draw the relevant right-angled triangle separately. Mixing a 3D sketch with the cosine rule without redrawing the flat triangle is the single biggest cause of errors in this type of question.
求点到直线的最短距离，或求直线与平面的夹角时，一定要把相关直角三角形单独重画出来。不重新画出平面三角形，就直接在三维草图上使用余弦定理，是这类题目最大的错误来源。
12. Common mistakes and a revision checklist | 常见错误与复习清单
Here are the errors that cost the most marks in IGCSE examinations on this topic.
以下是本主题在 IGCSE 考试中最容易失分的错误。
- Using the sine rule without a matching side-and-opposite-angle pair | 在没有"边与其对角"配对的情况下使用正弦定理
- Forgetting the ambiguous case when finding an angle with the sine rule | 用正弦定理求角时忘记检验两解情况
- Using the wrong angle as the included angle in the cosine rule | 在余弦定理中把错误的角当成夹角
- Losing the minus sign in −2bc cos A, or mishandling it when cos A is negative | 漏掉 −2bc cos A 的负号，或在 cos A 为负时处理错误
- Rounding to 3 significant figures in the middle of a multi-step calculation | 在多步计算中途就四舍五入到 3 位有效数字
- Leaving the calculator in radian mode when the question uses degrees | 题目使用角度制时计算器仍停留在弧度模式
- Forgetting units (cm, m, km) or giving area in the wrong squared units | 忘记写单位（cm、m、km），或面积的平方单位写错
- Using the area formula with a side and an angle that are not connected | 面积公式中使用的边与角并非相邻关系
A reliable revision routine is to redo one worked example of each type: sine rule for a side, sine rule for an angle with the ambiguous case, cosine rule for a side, cosine rule for an angle, and area with ½ab sin C. Then attempt two past-paper bearing questions and one 3D question.
可靠的复习流程是：把每种类型各重做一道例题——用正弦定理求边、用正弦定理求角（含两解检验）、用余弦定理求边、用余弦定理求角、以及用 ½ab sin C 求面积。然后再做两道历年真题中的方位角题和一道三维题。
an reproduce these from memory and label a diagram correctly, most questions on this topic become straightforward.
考试前，把四个关键公式
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