Cosine Rule and Its Typical Applications | 余弦定理及典型应用

📚 Cosine Rule and Its Typical Applications | 余弦定理及典型应用

The cosine rule (also called the law of cosines) relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a powerful tool for solving non-right-angled triangles, especially when we know two sides and the included angle, or all three sides.

余弦定理（又称余弦定律）将三角形的三边长度与某一个角的余弦值联系起来。它是解非直角三角形的重要工具，尤其适用于已知两边及其夹角，或已知三边的情形。

---

1. The Cosine Rule Formula | 余弦定理公式

For any triangle ABC, with sides a, b, c opposite angles A, B, C respectively, the cosine rule can be written in three equivalent forms:

对于任意三角形 ABC，设边 a、b、c 分别对应角 A、B、C，余弦定理有以下三种等价形式：

a² = b² + c² − 2bc·cos A

b² = a² + c² − 2ac·cos B

c² = a² + b² − 2ab·cos C

To find an angle from three sides, rearrange the formula:

若已知三边求角，可变形为：

cos A = (b² + c² − a²) / (2bc)

Here, the side on the left must be opposite the angle being calculated. Notice the pattern: the square of the target side equals the sum of squares of the other two sides minus twice their product times the cosine of the included angle.

注意，等式左边的边必须与所求角相对。规律是：所求边的平方等于另外两边平方之和减去这两边乘积的2倍与它们夹角余弦的乘积。

---

2. Derivation Using Coordinate Geometry | 利用坐标法推导

Place triangle ABC so that vertex C lies at the origin, and side CB lies along the positive x-axis. Let side CA have length b, side CB have length a, and the angle at C be γ. Then coordinates are A(b cos γ, b sin γ), B(a, 0). Using the distance formula between A and B:

将三角形 ABC 放置于坐标系中，使顶点 C 与原点重合，边 CB 沿 x 轴正方向。设边 CA 长为 b，边 CB 长为 a，角 C 为 γ。则 A 的坐标为 (b cos γ, b sin γ)，B 的坐标为 (a, 0)。利用两点间距离公式计算 AB 的长度：

c² = (b cos γ − a)² + (b sin γ)² = b² cos²γ − 2ab cos γ + a² + b² sin²γ

Since sin²γ + cos²γ = 1, this simplifies to c² = a² + b² − 2ab cos γ.

因为 sin²γ + cos²γ = 1，所以化简得 c² = a² + b² − 2ab cos γ。

---

3. Finding the Third Side Given Two Sides and the Included Angle | 已知两边及夹角求第三边

This is the most direct application of the cosine rule. Suppose a triangle has sides of length 7 cm and 10 cm, with an included angle of 60°. Then the third side x is found by:

这是余弦定理最直接的应用。例如，一个三角形有两条边分别为 7 cm 和 10 cm，它们的夹角为 60°，则第三边 x 可由下式求得：

x² = 7² + 10² − 2 × 7 × 10 × cos 60° = 49 + 100 − 70 = 79

x = √79 ≈ 8.89 cm

Always remember to take the positive square root because a side length must be positive. This method is essential when the triangle is not right-angled and the sine rule would require an extra step.

因为边长必须为正，所以只取正的平方根。当三角形不是直角三角形，而使用正弦定理需要额外步骤时，这种方法尤为重要。

---

4. Finding an Angle Given Three Sides | 已知三边求角

When all three sides are known, rearrange the cosine rule to find the cosine of the required angle, then use the inverse cosine function.

当已知所有三边长度时，可将余弦定理变形，先求出目标角的余弦值，再用反余弦函数求角。

Example: A triangle has sides a = 5, b = 6, c = 7. Find the largest angle, which is opposite the longest side c.

例：三角形三边 a = 5，b = 6，c = 7。求最大角，即最长边 c 所对的角。

cos C = (5² + 6² − 7²) / (2 × 5 × 6) = (25 + 36 − 49) / 60 = 12 / 60 = 0.2

C = cos⁻¹(0.2) ≈ 78.46°

The angle opposite the largest side is always the largest angle. Always verify that your answer is sensible: for a triangle, all angles must sum to 180°.

最大边所对的角一定是最大角。要检查答案是否合理：三角形内角和必须为180°。

---

5. Using the Cosine Rule to Determine Triangle Shape | 用余弦定理判断三角形形状

The sign of cos A reveals whether angle A is acute, right, or obtuse:

余弦值的符号可以判断角 A 是锐角、直角还是钝角：

• If cos A > 0, then A < 90° and the triangle is acute at A.

  若 cos A > 0，则 A < 90°，三角形在 A 处为锐角。

• If cos A = 0, then A = 90° and the triangle is right-angled at A.

  若 cos A = 0，则 A = 90°，三角形在 A 处为直角。

• If cos A < 0, then A > 90° and the triangle is obtuse at A.

  若 cos A < 0，则 A > 90°，三角形在 A 处为钝角。

For example, if a = 8, b = 15, c = 17, then a² + b² = 64 + 225 = 289 = c², so the triangle is right-angled. If a = 5, b = 6, c = 8, then c² = 64 > 5² + 6² = 61, so the angle opposite c is obtuse.

例如，若 a = 8, b = 15, c = 17，则 a² + b² = 64 + 225 = 289 = c²，所以该三角形为直角三角形。若 a = 5, b = 6, c = 8，则 c² = 64 > 5² + 6² = 61，所以 c 所对的角为钝角。

---

6. Relationship with the Pythagorean Theorem | 与勾股定理的关系

When the included angle is 90°, the cosine term vanishes, and the cosine rule reduces exactly to the Pythagorean theorem. In triangle ABC, if C = 90°, then cos C = 0, so:

当夹角为 90° 时，余弦项消失，余弦定理就退化为勾股定理。在三角形 ABC 中，若 C = 90°，则 cos C = 0，因此：

c² = a² + b²

This shows that the cosine rule is a generalisation of the Pythagorean theorem, valid for any triangle. Many examination questions ask students to recognise this special case without re-deriving it.

这说明余弦定理是勾股定理的推广，适用于任意三角形。许多考试题会要求考生直接识别这种特殊情况，而无需重新推导。

---

7. Typical Application: Measuring Distances | 典型应用：测量距离

The cosine rule is widely used in surveying and navigation. Suppose two observers stand 50 m apart at points A and B. They both look at a target T. At point A, the angle between AB and AT is 70°; at point B, the angle between BA and BT is 65°. To find the distance from A to T, first find ∠ATB = 180° − 70° − 65° = 45°. Then use the sine rule, or use the cosine rule directly if two sides and an included angle are known.

余弦定理广泛应用于测量和导航。假设两名观测者分别站在相距 50 m 的 A、B 两点，同时观察目标 T。在 A 点，AB 与 AT 的夹角为 70°；在 B 点，BA 与 BT 的夹角为 65°。要求 A 到 T 的距离，可先求 ∠ATB = 180° − 70° − 65° = 45°，然后使用正弦定理；如果已知两边及其夹角，则直接用余弦定理。

For a direct cosine rule application, imagine two ships leave the same port. Ship X sails 30 km due east, then turns 120° and sails 40 km. The distance from the port to the ship’s final position is found using the included angle of 120° (since the turn angle is measured from the original direction).

举一个直接应用余弦定理的例子：两艘船从同一港口出发。X 船先向正东航行 30 km，然后转弯 120° 再航行 40 km。此时港口到船最终位置的距离可利用夹角 120° 来求（因为转弯角是从原方向量得的）。

---

8. Typical Application: Resultant of Two Forces | 典型应用：两个力的合成

In physics, when two forces F₁ and F₂ act at an angle θ to each other, the magnitude of the resultant force R is given by the vector sum:

在物理中，当两个力 F₁ 和 F₂ 以夹角 θ 作用时，合力 R 的大小由矢量和给出：

R² = F₁² + F₂² + 2F₁F₂ cos θ

This formula is derived from the cosine rule, because the resultant vector forms a triangle with the two component vectors. Note the plus sign before the cosine term, since the included angle between the vectors is θ, but the triangle’s internal angle is 180° − θ, whose cosine is −cos θ.

该公式由余弦定理推导而来，因为合力矢量与两个分力矢量构成一个三角形。注意余弦项前是加号，因为两矢量之间的夹角为 θ，而三角形内角为 180° − θ，其余弦值为 −cos θ。

---

9. Solving Triangles with the Cosine Rule | 用余弦定理解三角形

To solve a triangle fully, you may need to combine the cosine rule with the sine rule and the angle sum property. A typical flow is:

要完整解一个三角形，可能需要将余弦定理、正弦定理以及内角和定理结合使用。典型流程为：

• Use the cosine rule to find the side opposite a known included angle.

  先用余弦定理求已知夹角的对边。

• Use the cosine rule again to find a second angle, or use the sine rule for speed.

  再用余弦定理求第二个角，或者用正弦定理加快速度。

• Subtract the two known angles from 180° to get the third angle.

  用180°减去两个已知角，得到第三个角。

When using the sine rule to find an angle, be careful of the ambiguous case (two possible angles). The cosine rule is safer because it gives a unique angle between 0° and 180°.

使用正弦定理求角时，要小心解的不确定性（可能有两个角满足条件）。余弦定理更安全，因为在0°到180°之间给出唯一确定的角。

---

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make the following errors when applying the cosine rule:

学生在应用余弦定理时常犯以下错误：

• Matching the wrong side to the wrong angle. Always label the triangle first, then choose the correct formula.

  将边与角对应错误。应先标注三角形，再选择正确的公式。

• Forgetting to take the square root when finding a side.

  求边长时忘记开平方根。

• Using degrees instead of radians (or vice versa) when evaluating cosine on a calculator.

  在计算器上求余弦时，混用角度制和弧度制。

• Rounding intermediate values too early, leading to inaccurate final answers.

  过早四舍五入中间值，导致最终答案不准确。

Exam tip: When a question gives two sides and a non-included angle, check whether the sine rule or cosine rule is more appropriate. The cosine rule is best when you have two sides and the included angle (SAS), or three sides (SSS).

考试技巧：当题目给出两边及一个非夹角时，判断使用正弦定理还是余弦定理更合适。如果已知两边及其夹角（SAS），或已知三边（SSS），余弦定理是最佳选择。

---

11. Worked Example: Mixed Use | 综合例题

In triangle ABC, AB = 8 cm, AC = 6 cm, and BC = 10 cm. Find the cosine of angle B.

在三角形 ABC 中，AB = 8 cm，AC = 6 cm，BC = 10 cm。求角 B 的余弦值。

Solution: Angle B is opposite side AC, so we use the formula with AC as the target side:

解：角 B 对边为 AC，因此以 AC 为目标边使用公式：

AC² = AB² + BC² − 2(AB)(BC) cos B

6² = 8² + 10² − 2 × 8 × 10 × cos B

36 = 164 − 160 cos B

cos B = (164 − 36) / 160 = 128 / 160 = 0.8

Thus B ≈ 36.87°. Notice that this triangle is not right-angled because 6² + 8² ≠ 10²; in fact 36 + 64 = 100, so it is right-angled. Wait, 36 + 64 = 100, so AC² + AB² = BC², meaning angle A is 90°, not angle B. The cosine rule confirms that cos B = 0.8.

因此 B ≈ 36.87°。注意，这个三角形并非以 B 为直角，因为 6² + 8² = 100 = 10²，所以角 A 是90°，而不是角 B。余弦定理验证了 cos B = 0.8。

---

12. Practice Problems | 练习与总结

Try the following problems to consolidate your understanding:

请尝试以下练习以巩固理解：

• In triangle PQR, PQ = 9 cm, QR = 12 cm, and ∠Q = 35°. Find PR.

  在三角形 PQR 中，PQ = 9 cm，QR = 12 cm，∠Q = 35°。求 PR。

• In triangle DEF, DE = 7 cm, EF = 11 cm, DF = 14 cm. Find the largest angle.

  在三角形 DEF 中，DE = 7 cm，EF = 11 cm，DF = 14 cm。求最大角。

• Two forces of 50 N and 80 N act at 75° to each other. Find the magnitude of the resultant.

  两个力分别为 50 N 和 80 N，夹角为 75°。求合力大小。

The cosine rule is one of the most versatile tools in trigonometry. Master its formula, recognise when to apply it, and practise the common pitfalls. With consistent practice, solving any non-right-angled triangle becomes straightforward.

余弦定理是三角学中最通用的工具之一。熟练掌握其公式，识别何时应用，并练习常见陷阱。通过持续练习，解任何非直角三角形都将变得简单明了。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）