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.
![How to Use the Law of Cosines to Find Angle Measure?](https://www.effortlessmath.com/wp-content/uploads/2023/01/Use-the-Law-of-Cosines-to-Find-Angle-Measure-512x240.jpg)
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 \(C\).
![This image has an empty alt attribute; its file name is answers.png](https://www.effortlessmath.com/wp-content/uploads/2019/12/answers.png)
- \(\color{blue}{123.94^{\circ \:\:}}\)
- \(\color{blue}{33.55^{\circ }}\)
Related to This Article
More math articles
- Top 5 Laptop Stands to Help You Teach Online
- 5th Grade OST Math FREE Sample Practice Questions
- Are knowledge checks mandatory on ALEKS?
- 6th Grade LEAP Math Worksheets: FREE & Printable
- 8th Grade SBAC Math FREE Sample Practice Questions
- CHSPE Math FREE Sample Practice Questions
- Math Assessment Test for College
- How to Find Constant of Proportionality?
- Top 10 Algebra 2 Textbooks in 2024 (Expert Recommendations)
- Getting Better at Math: Realistic Tips and Suggestions
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.