The Law of Cosines
The law of cosines or cosine rule is the relationship between the length of the sides of a triangle and the cosine of the angle formed by the triangle. Here, we will review the law of cosines.

The law of cosines describes the relationship between the length of the sides of a triangle and the cosine of the angle formed by the triangle. It is often referred to as the cosine rule.
Related Topics
- The Law of Sines
- How to Find Missing Sides and Angles of a Right Triangle
- How to Evaluate Trigonometric Function
- How to Solve Angles and Angle Measure
A step-by-step guide to the law of cosines
If the angles of an oblique triangle are measured by \(A\), \(B\), and \(C\) and \(a\), \(b\), and \(c\) are the lengths of the sides opposite the corresponding angles, then the square of one side of a triangle is equal to the total of the squares of the other two sides minus twice the product of the two sides plus the cosine of the included angle.
\(a^2=b^2+c^2-2bc .\cos A\)
\(b^2=a^2+c^2-2ac .\cos B\)
\(c^2=a^2+b^2-2ab .\cos C\)
Solving for the cosines gives the equivalent formulas:
\(cos A=\frac {b^2+c^2-a^2}{2bc}\)
\(cos B=\frac {c^2+a^2-b^2}{2ca}\)
\(cos C=\frac {a^2+b^2-c^2}{2ab}\)
The Law of Cosines – Example 1:
In the \(ABC\) triangle, find the remaining side.

To find side \(c\) use the law of cosines: \(c^2=a^2+b^2-2ab .\cos C\)
\(a=14, b=5, C=20\)
\(c^2\)\(=14^2+5^2-2(14)(5)(cos 20)=(196+25)-(140×cos 20)=221-(140×0.94)=221-131.6=89.4\)
\(c^2=89.4\) → \(c=\sqrt{89.44}= 9.45\)
The Law of Cosines – Example 2:
Find the angle \(B\) in the \(ABC\) triangle.

To find angle \(B\) use the law of cosines: \(cos B=\frac {c^2+a^2-b^2}{2ca}\)
\(b=20, a=8, c=14\)
\(cos B= \frac {14^2+8^2-20^2}{2(14)(8)} =\frac {196+ 64 – 400}{176}=\frac{-140}{224}=-0.625\)
Since \(cosB\) is negative, \(B\) is an obtuse angle.
\(B≅128.69 ^\circ \)
Exercises for the Law of Cosines
In the ABC triangle, find the side of c.
1.

2.

3.


- \(\color{blue}{31.12}\)
- \(\color{blue}{44.68}\)
- \(\color{blue}{21.49}\)
Related to This Article
More math articles
- Optimizing Betting Strategies in Crash Gambling with Mathematics
- How to Discover the Solutions: “HSPT Math for Beginners” Comprehensive Guide
- How long is the TEAS Test?
- How to Graph Absolute Value Function?
- Number Properties Puzzle – Challenge 15
- How to Prepare for the SSAT Middle-Level Math Test?
- The Wonderful World of the Triangle Inequality Theorem
- FREE 6th Grade MAP Math Practice Test
- CBEST Math Practice Test Questions
- The Best Books for Math Teachers
What people say about "The Law of Cosines - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.