The Law of Cosines is a rule for connecting the three side lengths of a triangle with one of its angles. It is especially useful when a triangle is not a right triangle, so the Pythagorean theorem does not directly apply. Students use it to solve triangles when they know two sides and the included angle, or when they know all three sides and need an angle.
It is a core tool in geometry, trigonometry, navigation, physics, and engineering.
Key Facts
- Main formula: c^2 = a^2 + b^2 - 2ab cos(C)
- Equivalent forms: a^2 = b^2 + c^2 - 2bc cos(A) and b^2 = a^2 + c^2 - 2ac cos(B)
- Use SAS when two sides and the included angle are known, then solve for the opposite side.
- Use SSS by rearranging: cos(C) = (a^2 + b^2 - c^2) / (2ab)
- If C = 90°, then cos(90°) = 0, so c^2 = a^2 + b^2.
- The largest side is always opposite the largest angle, which helps check if an answer is reasonable.
Vocabulary
- Law of Cosines
- A formula that relates the three sides of any triangle to the cosine of one angle.
- Included angle
- The angle formed between two known sides of a triangle.
- SAS
- A triangle information pattern where two sides and the included angle are known.
- SSS
- A triangle information pattern where all three side lengths are known.
- Opposite side
- The side across from a given angle in a triangle.
Common Mistakes to Avoid
- Using the wrong angle in the formula. The angle in c^2 = a^2 + b^2 - 2ab cos(C) must be the angle between sides a and b and opposite side c.
- Forgetting the negative sign before 2ab cos(C). This changes the relationship and can make an obtuse triangle look incorrectly small.
- Applying the Pythagorean theorem to every triangle. a^2 + b^2 = c^2 only works for right triangles, while the Law of Cosines works for any triangle.
- Rounding too early in multi-step problems. Early rounding can shift the final side or angle by a noticeable amount, so keep extra decimal places until the end.
Practice Questions
- 1 A triangle has a = 7, b = 10, and C = 60°. Find side c using the Law of Cosines.
- 2 A triangle has sides a = 8, b = 11, and c = 13. Find angle C to the nearest degree.
- 3 Explain why the Law of Cosines becomes the Pythagorean theorem when the included angle is 90°.