The Law of Cosines is a powerful rule for working with any triangle, not just right triangles. It connects the three side lengths of a triangle with the cosine of one angle. This makes it especially useful when a triangle is scalene and none of the angles are 90 degrees.
It matters because it lets you solve missing sides and angles in geometry, physics, surveying, navigation, and engineering.
Key Facts
- Law of Cosines for side a: a^2 = b^2 + c^2 - 2bc cos A
- Law of Cosines for side b: b^2 = a^2 + c^2 - 2ac cos B
- Law of Cosines for side c: c^2 = a^2 + b^2 - 2ab cos C
- To find an angle, rearrange: cos A = (b^2 + c^2 - a^2) / (2bc)
- If A = 90 degrees, then cos A = 0, so a^2 = b^2 + c^2, which is the Pythagorean Theorem
- Use the Law of Cosines when given SAS or SSS information in a triangle
Vocabulary
- Law of Cosines
- A formula that relates one side of a triangle to the other two sides and the cosine of the included angle.
- Included angle
- The angle formed between two known sides of a triangle.
- Scalene triangle
- A triangle in which all three side lengths are different.
- Opposite side
- The side across from a given angle in a triangle.
- SSS
- A triangle information pattern where all three side lengths are known.
Common Mistakes to Avoid
- Using the wrong opposite side label: side a must be opposite angle A, side b opposite angle B, and side c opposite angle C. Mixing labels leads to substituting values into the wrong formula.
- Forgetting the negative sign in -2bc cos A: the subtraction is part of the Law of Cosines. Changing it to addition gives an incorrect side length except in special unrelated cases.
- Using the Law of Sines for SAS information: SAS does not give an angle opposite a known side pair. The Law of Cosines is the correct first step when two sides and the included angle are known.
- Taking inverse cosine without checking calculator mode: angle answers depend on whether the calculator is in degrees or radians. Use degree mode when the problem gives angles in degrees.
Practice Questions
- 1 In triangle ABC, b = 7, c = 10, and A = 60 degrees. Use a^2 = b^2 + c^2 - 2bc cos A to find side a.
- 2 A triangle has side lengths a = 13, b = 14, and c = 15. Use cos A = (b^2 + c^2 - a^2) / (2bc) to find angle A to the nearest degree.
- 3 Explain why the Law of Cosines becomes the Pythagorean Theorem when the included angle is 90 degrees.