Law of Cosines
High School
Definition
Also known as the cosine rule. The law of cosines relates the cosine of an interior angle in a triangle to its sides. It helps us find aspects to a triangle such as its third side when we know two sides and the angle between them, or we can even find the angles inside a triangle if we have all three sides of the triangle.
Worked examples
\(c^2 = a^2 + b^2 - 2ab\cos C\)
The law of cosines formula: side \(c\) is opposite angle \(C\), and \(a\), \(b\) are the other two sides.
\(a = 5\), \(b = 7\), \(C = 60^\circ \)→\( c^2 = 25 + 49 - 2(5)(7)\cos 60^\circ = 74 - 35 = 39 \)→\( c = \sqrt{39}\)
Finding the third side when two sides and the included angle are known.
\(a = 8\), \(b = 6\), \(c = 10 \)→\( \cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{64 + 36 - 100}{96} = 0 \)→\( C = 90^\circ\)
Finding an angle when all three sides are known by rearranging the formula.
Common mistakes
- \(c^2 = a^2 + b^2 + 2ab\cos C\) → \(c^2 = a^2 + b^2 - 2ab\cos C\) The sign before \(2ab\cos C\) is negative, not positive.
- Using the angle not between the two known sides → Use the angle \(C\) that is between sides \(a\) and \(b\) to find side \(c\) The angle must be opposite the side you are finding.
- \(\cos C = \frac{c^2 - a^2 - b^2}{2ab}\) → \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\) When solving for the angle, rearrange carefully: move \(c^2\) to the right side with a negative sign.
Where you'll use it next
You'll apply the law of cosines when solving oblique triangles in trigonometry, physics problems involving forces and vectors, navigation and surveying, and later in calculus-based mechanics and engineering.
Found in 2 StudyPug lessons
Law of Cosines
11th Grade11thGrade 11 Math
The rule that relates the three sides of any triangle to one of its angles, for solving oblique triangles.
Applications of the Sine Rule and Cosine Rule
11th Grade11thGrade 11 Math
See how the sine rule and cosine rule solve triangles and real-life problems, from surveying distances to navigation, with clear worked steps.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026