余弦定理yúxián dìnglǐ

Core Concept

The Law of Cosines is an essential tool for solving triangles, describing the relationship between three sides and the cosine of an angle. It is a generalization of the Pythagorean theorem.

Theorem Statement

In $\triangle ABC$, let $a, b, c$ be the sides opposite to angles $A, B, C$ respectively:

$a^2 = b^2 + c^2 - 2bc\cos A$ $b^2 = a^2 + c^2 - 2ac\cos B$ $c^2 = a^2 + b^2 - 2ab\cos C$

Corollary (solving for angles): $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Applications

1. Given Two Sides and Included Angle (SAS)

Example: In $\triangle ABC$, $b = 3$, $c = 4$, $A = 60°$, find $a$.

Solution: $a^2 = 9 + 16 - 2(3)(4)\cos 60° = 25 - 12 = 13$ $a = \sqrt{13}$

2. Given Three Sides (SSS), Find Angles

Example: In $\triangle ABC$, $a = 5$, $b = 7$, $c = 8$, find $\cos A$.

Solution: $\cos A = \frac{49 + 64 - 25}{112} = \frac{11}{14}$

Relationship with Pythagorean Theorem

When $A = 90°$, $\cos A = 0$: $a^2 = b^2 + c^2$

This is the Pythagorean theorem. Thus, the Law of Cosines generalizes the Pythagorean theorem.

Common Misconceptions

❌ Misconception 1: Wrong sign in formula

Wrong: $a^2 = b^2 + c^2 + 2bc\cos A$

Correct: $a^2 = b^2 + c^2 - 2bc\cos A$ (minus sign)

Study Tips

1. ✅ Understand essence: Law of Cosines generalizes Pythagorean theorem
2. ✅ Memorize formula: $a^2 = b^2 + c^2 - 2bc\cos A$
3. ✅ Master applications: SAS and SSS cases
4. ✅ Combine with Law of Sines: Use flexibly

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💡 Exam Tip: Law of Cosines is as important as Law of Sines, mandatory in CSCA exams!