Solving Non-Right Triangles: Sine Rule and Cosine Rule | 非直角三角形求解：正弦定理与余弦定理

📚 Solving Non-Right Triangles: Sine Rule and Cosine Rule | 非直角三角形求解：正弦定理与余弦定理

For right-angled triangles, the trigonometric ratios sine, cosine and tangent are sufficient to find unknown sides and angles. But what happens when a triangle has no right angle? We need two powerful tools — the Sine Rule and the Cosine Rule — which extend trigonometry to any triangle.

对于直角三角形，正弦、余弦和正切这三种三角比足以求出未知的边和角。但如果三角形没有直角该怎么办？我们需要两个强大的工具——正弦定理和余弦定理——它们将三角学推广到了任意三角形。

1. When Do We Need the Sine and Cosine Rules? | 何时需要正弦定理与余弦定理

Any triangle can be solved (i.e., all sides and angles found) if we know three of its six parts, with at least one side. The choice of rule depends on which three parts are given.

任何三角形，只要已知六个要素中的三个（其中至少有一条边），就可以求解出全部边和角。选择哪个定理取决于已知的是哪三个要素。

If we know two angles and one side (AAS or ASA), we use the Sine Rule.

如果已知两角一边（AAS 或 ASA），使用正弦定理。
If we know two sides and a non-included angle (SSA), the Sine Rule applies — but this case can be ambiguous.

如果已知两边及其中一边的对角（SSA），正弦定理适用——但这种情况可能存在“两解”的歧义。
If we know two sides and the included angle (SAS), or three sides (SSS), we use the Cosine Rule.

如果已知两边及其夹角（SAS），或三边（SSS），使用余弦定理。

In short: the Sine Rule works with angle–side pairs; the Cosine Rule works with side-side-side or side-angle-side.

简而言之：正弦定理处理“对角对边”的组合；余弦定理处理“边边边”或“边角边”的组合。

2. The Sine Rule (Law of Sines) | 正弦定理

For any triangle ABC with sides a, b, c opposite angles A, B, C respectively:

对于任意三角形 ABC，边 a、b、c 分别对角 A、B、C：

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

This elegant relation says that the ratio of a side to the sine of its opposite angle is constant throughout the triangle.

这一简洁的关系说明：三角形中每条边与其对角正弦之比在整个三角形中保持不变。

Equivalently, we may write it in reciprocal form:

等价地，也可以写成倒数形式：

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

The Sine Rule is used when we know:

正弦定理适用于已知：

two angles and any side (AAS or ASA), or
two sides and an angle opposite one of them (SSA).

两角及任意一边（AAS 或 ASA）；
两边及其中一边的对角（SSA）。

3. Derivation of the Sine Rule | 正弦定理的推导

The proof uses only the definition of sine in right triangles formed by an altitude.

推导只需利用“作高”得到的直角三角形中正弦的定义。

Draw altitude h from C to side AB (or its extension). In the right triangles formed:

从顶点 C 向 AB 边（或其延长线）作高 h。在形成的两个直角三角形中：

sin A = h / b , sin B = h / a

Therefore h = b sin A = a sin B, which gives a / sin A = b / sin B. Similarly, repeating the argument with another altitude gives the full statement.

因此 h = b sin A = a sin B，从而得到 a / sin A = b / sin B。类似地，换一条高重复论证，便得到完整的定理。

This derivation also shows why the sine of an obtuse angle works: the altitude may fall outside the triangle, but the sine ratio remains valid because sin(180° − θ) = sin θ.

该推导还说明了为什么钝角的正弦也适用：高可能落在三角形外部，但由于 sin(180° − θ) = sin θ，正弦比值依然成立。

4. The Cosine Rule (Law of Cosines) | 余弦定理

For any triangle ABC:

对于任意三角形 ABC：

a² = b² + c² − 2bc · cos A

b² = a² + c² − 2ac · cos B

c² = a² + b² − 2ab · cos C

Notice the pattern: each side squared equals the sum of the squares of the other two sides, minus twice their product times the cosine of the included angle.

注意规律：任意一边的平方，等于另外两边平方之和，减去这两边乘积的 2 倍再乘以它们夹角的余弦。

The Cosine Rule is used when we know:

余弦定理适用于已知：

two sides and the included angle (SAS), or
three sides (SSS).

两边及其夹角（SAS）；
三边（SSS）。

When finding an angle from three sides, rearrange:

当用三边求角时，可变形为：

cos A = (b² + c² − a²) / (2bc)

5. Derivation of the Cosine Rule | 余弦定理的推导

Place triangle ABC with C at the origin, side b along the x-axis. Then B has coordinates (a, 0) and A has coordinates (b cos C, b sin C).

将三角形 ABC 的顶点 C 放在原点，边 b 沿 x 轴方向。于是 B 的坐标为 (a, 0)，A 的坐标为 (b cos C, b sin C)。

The distance between A and B is c. Using the distance formula:

A 与 B 之间的距离为 c。利用距离公式：

c² = (b cos C − a)² + (b sin C)²

Expanding:

展开：

c² = b² cos² C − 2ab cos C + a² + b² sin² C

Since cos² C + sin² C = 1:

因为 cos² C + sin² C = 1：

c² = a² + b² − 2ab cos C

This beautiful coordinate proof is concise and also shows the Pythagorean theorem as a special case when C = 90°.

这一简洁的坐标法证明同时揭示：当 C = 90° 时，余弦定理就退化为勾股定理。

6. Applying the Sine Rule: Worked Example | 正弦定理的应用：例题

Example 1: In triangle ABC, angle A = 48°, angle B = 65°, and side a = 12 cm. Find side b.

例 1：在三角形 ABC 中，A = 48°，B = 65°，边 a = 12 cm。求边 b。

First find the missing angle: C = 180° − 48° − 65° = 67°. Using the Sine Rule:

先求第三个角：C = 180° − 48° − 65° = 67°。使用正弦定理：

a / sin A = b / sin B

12 / sin 48° = b / sin 65°

b = 12 × sin 65° / sin 48° ≈ 12 × 0.9063 / 0.7431 ≈ 14.6 cm

Notice that we used only the relevant pair of ratios; we did not need angle C to find b, but it is often useful for subsequent steps.

注意我们只用了相关的两对比例；求 b 并不需要 C，但在后续步骤中通常有用。

7. The Ambiguous Case (SSA) | 两边一对角（SSA）的歧义情况

When we know two sides and an angle opposite one of them, the Sine Rule may give zero, one, or two possible triangles.

当已知两边及其中一边的对角时，正弦定理可能给出零个、一个或两个可能的三角形。

This is known as the ambiguous case. For example, in triangle ABC, suppose we know a, b and angle A.

这就是所谓的“歧义情况”。例如在三角形 ABC 中，已知 a、b 和角 A。

Solving sin B = b sin A / a, we may find:

通过解 sin B = b sin A / a，我们可能得到：

sin B > 1: no such triangle exists (the side a is too short).

sin B > 1：三角形不存在（a 太短）。
sin B = 1: one unique right triangle (B = 90°).

sin B = 1：唯一的直角三角形（B = 90°）。
sin B < 1: two possible angles B₁ = θ and B₂ = 180° − θ; both may be valid if the corresponding angle C is positive.

sin B < 1：两个可能的角 B₁ = θ 和 B₂ = 180° − θ；只要对应的角 C 为正，两者都可能成立。

Example 2: Given a = 7 cm, b = 9 cm, and angle A = 35°, find angle B.

例 2：已知 a = 7 cm，b = 9 cm，A = 35°，求角 B。

sin B = b sin A / a = 9 sin 35° / 7 ≈ 9 × 0.5736 / 7 ≈ 0.7375

Thus B₁ = arcsin(0.7375) ≈ 47.5°, and B₂ = 180° − 47.5° = 132.5°. Both are possible because C = 180° − 35° − 47.5° = 97.5° > 0 and C = 180° − 35° − 132.5° = 12.5° > 0.

因此 B₁ ≈ 47.5°，B₂ = 180° − 47.5° = 132.5°。两者都可行，因为对应角 C 分别为 97.5° 和 12.5°，均大于 0°。

Always check whether the obtuse solution is geometrically possible before accepting both.

在同时接受两个解之前，务必检查钝角解在几何上是否可行。

8. Applying the Cosine Rule: Worked Examples | 余弦定理的应用：例题

Example 3 (SAS): In triangle ABC, b = 8 cm, c = 11 cm, and angle A = 52°. Find side a.

例 3（边角边）：在三角形 ABC 中，b = 8 cm，c = 11 cm，A = 52°。求边 a。

a² = b² + c² − 2bc cos A = 8² + 11² − 2 × 8 × 11 × cos 52°

a² = 64 + 121 − 176 × 0.6157 ≈ 185 − 108.36 ≈ 76.64

a ≈ √76.64 ≈ 8.75 cm

Example 4 (SSS): A triangle has sides 6 cm, 7 cm, 8 cm. Find the largest angle (opposite the side of 8 cm).

例 4（边边边）：三角形的三边长为 6 cm、7 cm、8 cm。求最大角（对着 8 cm 的边）。

cos C = (a² + b² − c²) / (2ab) = (6² + 7² − 8²) / (2 × 6 × 7)

cos C = (36 + 49 − 64) / 84 = 21 / 84 = 0.25

C = arccos(0.25) ≈ 75.5°

The Cosine Rule is particularly suited for finding angles when all three sides are known, because the sine rule would require an extra step to avoid ambiguity.

当三边已知时，余弦定理特别适合求角，因为用正弦定理求角需要额外步骤并警惕两解歧义。

9. Area of a Non-Right Triangle | 非直角三角形的面积

Closely related to these rules is the area formula:

与这两个定理密切相关的是面积公式：

Area = ½ · ab · sin C =\frac{1}{2}ab sin C

Area = ½ × a × b × sin C

In words: the area of any triangle equals half the product of two sides times the sine of the included angle.

也就是说：任意三角形的面积等于两边之积的一半，再乘以它们夹角的正弦。

Example 5: Two sides of a triangle are 10 cm and 14 cm with included angle 40°. Find the area.

例 5：三角形的两边为 10 cm 和 14 cm，夹角为 40°。求面积。

Area = ½ × 10 × 14 × sin 40° = 70 × 0.6428 ≈ 45.0 cm²

This formula is useful in mechanics, physics and geometry when two sides and an included angle are easier to measure than a perpendicular height.

当两边及其夹角比垂线高度更容易测量时，这个公式在力学、物理和几何中都很有用。

10. Choosing the Right Rule: A Decision Strategy | 选择正确的定理：决策策略

The table below summarizes which rule to use depending on the given information.

下表总结了根据已知信息应选择哪个定理。

已知条件 Given	首选定理 First Choice	备注 Remark
两角一边 AAS / ASA	正弦定理 Sine Rule	先求第三个角，再求其他边 Find third angle first, then sides
两边一对角 SSA	正弦定理 Sine Rule	⚠ 警惕两解 ambiguity risk
两边及其夹角 SAS	余弦定理 Cosine Rule	直接求第三边 find third side directly
三边 SSS	余弦定理 Cosine Rule	先求最大角 avoid ambiguity

When in doubt, remember the triangle angle sum: A + B + C = 180°. This alone often decides whether a second solution is valid.

若犹豫不决，请记住三角形的内角和：A + B + C = 180°。仅这一条就常能判定第二个解是否有效。

11. Common Mistakes and Pitfalls | 常见错误与易错点

Using the Sine Rule in an SAS situation: the given angle is not opposite a known side, so the rule cannot be applied directly.

在“边角边”情形误用正弦定理：所给角不对着已知边，无法直接套用。
Ignoring the ambiguous case and discarding the obtuse angle without checking if C > 0.

忽略两解情况，不检查 C > 0 就舍弃钝角解。
Switching sides and angles: ensure that a is opposite A, b opposite B, etc.

张冠李戴：务必保证 a 对着 A，b 对着 B，依此类推。
Rounding too early: keep at least 3 significant figures during intermediate steps.

过早四舍五入：中间步骤请至少保留 3 位有效数字。
Forgetting to use the inverse sine (arcsin) or inverse cosine (arccos) when solving for an angle.

求角时忘记使用反正弦（arcsin）或反余弦（arccos）。

12. Complete Worked Example: Solving a Whole Triangle | 综合例题：求解整个三角形

Example 6: In triangle PQR, p = 15 cm, q = 20 cm, and angle P = 30°. Solve the triangle fully.

例 6：在三角形 PQR 中，p = 15 cm，q = 20 cm，角 P = 30°。完整地解这个三角形。

Step 1: Use the Sine Rule to find angle Q.

第一步：用正弦定理求角 Q。

sin Q = q sin P / p = 20 sin 30° / 15 = 20 × 0.5 / 15 ≈ 0.6667

Q₁ = arcsin(0.6667) ≈ 41.8°. The alternate solution Q₂ = 180° − 41.8° = 138.2° gives R = 180° − 30° − 138.2° = 11.8°, which is valid. So two triangles are possible.

Q₁ ≈ 41.8°。另一个解 Q₂ = 180° − 41.8° = 138.2°，相应的 R = 11.8°，同样有效。因此这里有两个可能的三角形。

Step 2: For each case, find side r using the Sine Rule.

第二步：在每种情形下，用正弦定理求边 r。

Case 1: Q = 41.8°, R = 180° − 30° − 41.8° = 108.2°:

情形 1：Q = 41.8°，R = 108.2°：

r = p sin R / sin P = 15 × sin 108.2° / sin 30° ≈ 15 × 0.9500 / 0.5 ≈ 28.5 cm

Case 2: Q = 138.2°, R = 11.8°:

情形 2：Q = 138.2°，R = 11.8°：

r = 15 × sin 11.8° / sin 30° ≈ 15 × 0.2045 / 0.5 ≈ 6.1 cm

Step 3: Verify with the angle sum and, if possible, the Cosine Rule on one of the results.

第三步：用内角和验证；如有可能，再用余弦定理验证其中一个结果。

This example demonstrates exactly why the SSA case is called ambiguous: the given data can describe two different triangles.

这个例子精确说明了为什么 SSA 情形被称为“歧义”：同样的数据可能描述两个不同的三角形。

13. Summary and Examination Tips | 总结与考试建议

Mastering the Sine Rule and Cosine Rule gives you a complete toolkit for solving any triangle, including those with obtuse angles.

熟练掌握正弦定理与余弦定理，你便拥有了求解任何三角形（包括含钝角的三角形）的完整工具包。

Identify the given information type first: AAS, SSA, SAS, or SSS.

先判断已知信息类型：AAS、SSA、SAS 或 SSS。
Use the Sine Rule when an angle and its opposite side are paired; use the Cosine Rule when sides dominate.

当角与对边成对出现时，用正弦定理；当以边为主时，用余弦定理。
In SSA cases, always examine the possible values of the unknown angle; ensure the third angle is positive.

在 SSA 情形中，始终检查未知角的可能取值；确保第三个角为正。
Remember the area formula: Area = ½ · ab · sin C — it often pairs with these rules.

记住面积公式：Area = ½ × a × b × sin C——它经常与这两个定理一并使用。
Draw a clear diagram and label sides and angles correctly before applying any formula.

先画清楚示意图并正确标注边和角，再套用任何公式。
On IB exams, state which rule you are using; method marks are awarded for clear reasoning.

在 IB 考试中，写明你在用哪个定理；清晰的推理过程会得到步骤分。

With practice, deciding between the Sine Rule and the Cosine Rule becomes automatic, and the ambiguous case loses its mystery.

多加练习后，在正弦定理与余弦定理之间做选择会变成一种本能，两解情况也不再神秘。

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Solving Non-Right Triangles: Sine Rule and Cosine Rule | 非直角三角形求解：正弦定理与余弦定理