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.
Understanding Geometry: The Law of Cosines in Depth
Each angle has a matching opposite side. In standard triangle labels, angle A faces side a, angle B faces side b, and angle C faces side c. This matching is not a decoration.
It tells you which angle belongs in a calculation for a particular missing side. Students often lose points by using the angle beside the target side instead of the angle across from it.
Mark the opposite side before doing any arithmetic. It keeps the labels organized, especially when a diagram is tilted or drawn without equal-looking sides.
The cosine part measures how much one known side points in the direction of the other. Imagine dropping a perpendicular from one vertex to form two right triangles inside the original triangle. The horizontal part of a side is its projection, and cosine gives the size of that projection.
This is why the rule contains a correction based on the included angle. For an acute angle, the correction reduces the square of the opposite side.
For an obtuse angle, cosine is negative, so subtracting that negative value makes the opposite side longer. A wider angle must face a longer side.
With SAS information, two side lengths and the angle between them are known. Start by identifying the side opposite the given angle. That is normally the side to find.
Substitute the two adjacent side lengths and the included angle into the matching version of the rule. Keep extra decimal places while calculating cosine and while squaring. Round only after taking the square root at the end.
A useful estimate comes first. If the included angle is close to zero degrees, the missing side should be close to the difference of the known sides. If it is close to 180 degrees, the missing side should be close to their sum.
With SSS information, all three side lengths are known and an angle is missing. Choose the angle you need, then place its opposite side as the side being compared with the other two. Rearranging gives a cosine value.
Use the inverse cosine button on a calculator to turn that value into an angle. Check that the calculator is in degree mode unless the problem specifically uses radians.
Because measurements may be rounded, a result slightly above one or slightly below negative one usually signals an entry error or excessive rounding. The three final angles should add to 180 degrees.
This reasoning appears whenever distances are measured indirectly. A surveyor can use two measured paths and the turn between them to find the distance across land. A robot can estimate its position after moving in two directions.
In physics, two displacement vectors form a triangle, and their angle changes the size of the combined displacement. When learning the method, draw a clear sketch, label opposite pairs, and decide whether the given angle is included.
Then use size facts as a check. The largest side faces the largest angle, and a triangle cannot have a side length greater than or equal to the sum of the other two.
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.