How to Use the Law of Cosines to Find Angle Measure?
If we know the sizes of the three sides of the triangle, we can use the law of cosines to find the size of each angle of the triangle. In this guide, you will learn more about the law of cosines.

Step-by-step guide to using the law of cosines to find angle measure
The law of cosine says that the square of each side of a triangle is equal to the difference between the sum of squares of the other two sides and twice the product of other sides and the cosine angle included between them.
Let \(a, b,\) and \(c\) be the lengths of the three sides of a triangle and \(A, B,\) and \(C\) be the three angles of the triangle. Then, the law of cosine states that:
- \(\color{blue}{a^2=b^2+c^2-2bc.\:cos\:A}\)
- \(\color{blue}{b^2=\:c^2\:+\:a^2\:-\:2ca·\:cosB}\)
- \(\color{blue}{c^2=\:a^2+\:b^2-\:2ab·\:cosC}\)
If we know the sizes of the three sides of the triangle, we can use the law of cosines to find the size of each angle of the triangle. These formulas can be used to find the cosine of any angle of \(∆\: ABC\):
- \(\color{blue}{cos\:A=\frac{\:b^2+c^2-a^2}{2bc}}\)
- \(\color{blue}{cos\:B=\:\frac{a^2+c^2-b^2}{2ac}}\)
- \(\color{blue}{cos\:C=\:\frac{a^2+b^2-c^2}{2ab}}\)
Using the Law of Cosines to Find Angle Measure – Example 1:
In \(ABC\) triangle, \(a=12,\:b=8,\:c=6\). Find the angle \(B\).
Solution:
Write the law of cosines in terms of \(cos B\): \(cos\:B=\:\frac{a^2+c^2-b^2}{2ac}\)
\(cos\:B=\frac{12^2+6^2-8^2}{2\times 12\times 6}\)
\(cos B =0.8\)
\(B= 36.33^{\circ }\)
Exercises for Using the Law of Cosines to Find Angle Measure
- In \(∆\:ABC\), \(a=25,\:b=10,\:c=18\). Find the angle \(A\).
- In \(∆\:ABC\), \(a=9,\:b=8,\:c=5\). Find the angle \(C\).

- \(\color{blue}{123.94^{\circ \:\:}}\)
- \(\color{blue}{33.55^{\circ }}\)
Related to This Article
More math articles
- Introduction to Sets
- Math Topics You Need to Learn in Medicine
- Math Classroom Decorations and Bulletin Board Supplies
- How to Find the Area of a Quarter Circle?
- 3rd Grade M-STEP Math Worksheets: FREE & Printable
- Ratio, Proportion and Percentages Puzzle – Critical Thinking 7
- CLEP College Algebra Worksheets: FREE & Printable
- How to Prepare for the ParaPro Math Test?
- How to Categorize Rational Numbers
- 6th Grade SBAC Math FREE Sample Practice Questions
What people say about "How to Use the Law of Cosines to Find Angle Measure? - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.