The Sine and Cosine Rules: Solving Triangles | 正弦定理与余弦定理：解三角形

📚 The Sine and Cosine Rules: Solving Triangles | 正弦定理与余弦定理：解三角形

Trigonometry is a key topic in IGCSE Mathematics, and the sine rule and cosine rule are two powerful formulas used to solve non-right-angled triangles. Once you know when and how to apply them, many exam questions become straightforward.

三角学是 IGCSE 数学中的重点内容。正弦定理与余弦定理是解非直角三角形的两大有力工具。只要掌握它们的适用条件和正确用法，很多考试题都会变得简单明了。

1. What Are the Two Rules? | 两大定理是什么

The sine rule connects a side to the sine of its opposite angle. The cosine rule connects a side to the other two sides and the included angle. Both rules work for any triangle, not just a right-angled triangle.

正弦定理将一条边与其对角的正弦值联系起来；余弦定理则将一条边与另外两条边及其夹角联系起来。两大定理对任意三角形都适用，不限于直角三角形。

Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
当你已知两角一边，或两边及其中的一对角时，应使用正弦定理。
Use the cosine rule when you know two sides and the included angle, or all three sides.
当你已知两边及其夹角，或已知三边时，应使用余弦定理。

2. The Sine Rule: Formula | 正弦定理：公式

For any triangle ABC, label sides a, b and c so that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The sine rule states that the ratio of each side to the sine of its opposite angle is constant.

对任意三角形 ABC，设边 a 与角 A 相对，边 b 与角 B 相对，边 c 与角 C 相对。正弦定理说明：每条边与其对角正弦值的比是一个常数。

a / sin A = b / sin B = c / sin C

This single equation can be rearranged to find either a missing side or a missing angle. When calculating an angle, it is often easier to use the reciprocal form sin A / a = sin B / b.

这个等式可以通过变形来求缺失的边或角。当需要求角时，使用倒数的形式 sin A / a = sin B / b 通常更为方便。

3. Using the Sine Rule to Find a Side | 用正弦定理求边

Suppose in triangle ABC you know angle A = 50°, angle B = 72° and side a = 12 cm. To find side b, write the sine rule in the form b / sin B = a / sin A.

假设在三角形 ABC 中，角 A = 50°，角 B = 72°，边 a = 12 cm。要求边 b，可将正弦定理写成 b / sin B = a / sin A 的形式。

b = 12 × sin 72° ÷ sin 50° = 14.9 cm

Always check that your calculator is in degree mode, not radian mode, before evaluating sine values.

在计算正弦值之前，务必检查计算器处于角度模式（Degree Mode），而不是弧度模式（Radian Mode）。

4. Using the Sine Rule to Find an Angle | 用正弦定理求角

To find an angle, use the reciprocal form. If b = 15, c = 10 and angle B = 80°, then sin C can be found from c / sin C = b / sin B.

求角时使用倒数形式。若 b = 15，c = 10，角 B = 80°，则由 c / sin C = b / sin B 可求出 sin C。

sin C = 10 × sin 80° ÷ 15 = 0.657

Hence C = arcsin(0.657) = 41.0° (to 1 d.p.). Remember that the inverse sine function gives a principal value, but another possible angle exists; this leads to the ambiguous case.

因此 C = arcsin(0.657) = 41.0°（精确到一位小数）。注意反三角函数给出一个主值，但还可能存在另一个符合条件的角，这就引出了“两解情况”。

5. The Ambiguous Case | 两解情况

When you know two sides and a non-included angle (SSA), the sine rule may produce two different triangles. For example, if sin C = 0.657, then C could be 41.0° or 139.0°, because sin(180° − 41°) = sin 41°.

当你已知两边及其中一边对角（SSA）时，正弦定理可能产生两个不同的三角形。例如，若 sin C = 0.657，则 C 可能是 41.0° 或 139.0°，因为 sin(180° − 41°) = sin 41°。

If the obtuse option makes the angle sum exceed 180°, discard it.
如果钝角选项会使三角形内角和超过 180°，就应舍去。
If both options are valid, give both answers clearly.
如果两个选项都满足条件，则需要明确写出两个答案。
This situation only arises with SSA, never with ASA or AAS.
这种情况只会在 SSA 中出现，而不会出现在 ASA 或 AAS 中。

6. The Cosine Rule: Formula | 余弦定理：公式

The cosine rule is used when you have two sides and the included angle (SAS) or three sides (SSS). For a triangle with sides a, b and c, and angle A opposite side a, the rule is as follows.

余弦定理适用于已知两边及其夹角（SAS）或已知三边（SSS）的情形。对于边 a、b、c，角 A 与边 a 相对，规则如下。

a² = b² + c² − 2bc cos A

By relabelling the triangle, you can also write b² = a² + c² − 2ac cos B and c² = a² + b² − 2ab cos C.

通过重新标注三角形，也可写为 b² = a² + c² − 2ac cos B 和 c² = a² + b² − 2ab cos C。

7. Using the Cosine Rule to Find a Side | 用余弦定理求边

Suppose b = 8 cm, c = 11 cm and angle A = 35°. To find side a, substitute into the cosine rule.

设 b = 8 cm，c = 11 cm，角 A = 35°。要求边 a，代入余弦定理即可。

a² = 8² + 11² − 2 × 8 × 11 × cos 35°

a² = 64 + 121 − 144.1 = 40.9

a = √40.9 = 6.39 cm

This method is reliable because the cosine rule always gives a unique value for a side when the included angle is known.

这种方法非常可靠，因为当夹角已知时，余弦定理求边只会得到唯一值。

8. Using the Cosine Rule to Find an Angle | 用余弦定理求角

Rearrange the cosine rule to make cos A the subject. This form is useful when all three sides are known.

将余弦定理变形，用其他量表示 cos A。当三边都已知时，这个形式非常有用。

cos A = (b² + c² − a²) ÷ (2bc)

Example: a = 7, b = 9, c = 12. Then cos A = (9² + 12² − 7²) ÷ (2 × 9 × 12) = 176 ÷ 216 = 0.8148, so A = 35.4°.

例：a = 7，b = 9，c = 12。则 cos A = (9² + 12² − 7²) ÷ (2 × 9 × 12) = 176 ÷ 216 = 0.8148，因此 A = 35.4°。

The inverse cosine function gives a unique angle between 0° and 180°, so there is no ambiguous case with the cosine rule.

反余弦函数在 0° 到 180° 之间给出唯一角，因此余弦定理不存在两解情况。

9. Area of a Triangle Using Trigonometry | 用三角函数求三角形面积

For any triangle, if you know two sides and the included angle, the area can be found using the formula below. This is especially useful in non-right-angled triangles where the perpendicular height is not given.

对任意三角形，若已知两边及其夹角，可用下面这个公式求面积。对于没有给出垂直高度的非直角三角形，这个公式尤其方便。

Area = ½ ab sin C

Here, a and b are two sides, and C is the angle between them. Make sure the angle is measured in degrees, and give your final answer in square units.

其中 a 和 b 是两条边，C 是它们的夹角。确保角度使用度数表示，最终答案要使用适当的面积单位。

10. Which Rule Should I Use? | 如何选择合适定理

Exam questions often require you to identify the most efficient method. The table below summarises the choice based on the information given.

考试题目经常要求你判断最有效的解题方法。下表总结了根据已知条件应如何选择。

Given Information	Rule to Use
Two angles and one side (ASA or AAS)	Sine Rule
Two sides and a non-included angle (SSA)	Sine Rule, check ambiguous case
Two sides and the included angle (SAS)	Cosine Rule for the third side
All three sides (SSS)	Cosine Rule for an angle

11. Worked Example: Mixed Problem | 综合例题

A triangular garden has sides 8 m, 11 m and 14 m. Find the largest angle and the area of the garden. The largest angle is opposite the longest side, which is 14 m.

一块三角形花园的三边长分别为 8 m、11 m 和 14 m。求最大角以及花园面积。最大角对着最长边 14 m。

cos A = (8² + 11² − 14²) ÷ (2 × 8 × 11) = (64 + 121 − 196) ÷ 176 = −11 ÷ 176 = −0.0625

A = cos⁻¹(−0.0625) = 93.6°

To find the area, use two sides and the included angle: Area = ½ × 8 × 11 × sin 93.6° ≈ 43.9 m². The largest angle is 93.6°, so this is an obtuse triangle.

求面积时，用两边及其夹角：Area = ½ × 8 × 11 × sin 93.6° ≈ 43.9 m²。最大角为 93.6°，因此这是钝角三角形。

12. Common Mistakes and Checklist | 常见错误与检查清单

The most common errors in IGCSE trigonometry questions come from careless use of the calculator or mixing up the formulas. Follow the checklist below to stay accurate.

IGCSE 三角学题目中最常见的错误来自计算器使用不当或混淆公式。请按照下面的清单来确保准确性。

Always use degree mode on your calculator.
始终将计算器设置为角度模式。
Label the triangle clearly before substituting values.
代入数值前先清楚标注三角形的边和角。
Check whether the sine rule has two possible answers.
检查正弦定理是否可能存在两个答案。
Use the cosine rule for SAS and SSS, not the sine rule.
在处理 SAS 和 SSS 时使用余弦定理，而不是正弦定理。
Round only at the final answer, not during intermediate steps.
只在最后答案处四舍五入，不要在中间步骤中过早取近似值。

Once you are comfortable with these two rules, solving triangle problems becomes a systematic process. Practice identifying the correct rule quickly, and always sketch the triangle before starting.

当你熟练掌握这两个定理后，解三角形问题将变成一个系统化的过程。练习快速判断应使用的定理，并在做题前先画出三角形草图。

Published by TutorHao | 数学 Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

The Sine and Cosine Rules: Solving Triangles | 正弦定理与余弦定理：解三角形