How to Use the Law of Cosines to Find Angle Measure?

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 the 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 cosines 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

1. In \(∆\:ABC\), \(a=25,\:b=10,\:c=18\). Find the angle \(A\).
2. In \(∆\:ABC\), \(a=9,\:b=8,\:c=5\). Find the angle \(C\).

1. \(\color{blue}{123.94^{\circ \:\:}}\)
2. \(\color{blue}{33.55^{\circ }}\)