Law of Cosines

Definition

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful for solving triangles that are not right-angled.

5 Must Know Facts For Your Next Test

1. The Law of Cosines formula is $c^2 = a^2 + b^2 - 2ab\cos(C)$, where \(a\), \(b\), and \(c\) are the sides, and \(C\) is the angle opposite side \(c\).
2. It generalizes the Pythagorean theorem, which is a special case when the angle is $90^\circ$.
3. The formula can be rearranged to solve for an angle: $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.
4. It can also be written in terms of other angles: $a^2 = b^2 + c^2 - 2bc\cos(A)$ and $b^2 = a^2 + c^2 - 2ac\cos(B)$.
5. The Law of Cosines is used to find unknown sides or angles in any triangle, especially when two sides and an included angle or all three sides are known.

Review Questions

• Write down the Law of Cosines formula and explain each part.
• How does the Law of Cosines relate to the Pythagorean theorem?
• Given sides \(a = 7\), \(b = 10\), and angle \(C = 45^\circ\), use the Law of Cosines to find side \(c\).

Related terms

Law of Sines:

A trigonometric law that relates the lengths of sides of a triangle to sines of its angles: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Pythagorean Theorem:

A fundamental relation in Euclidean geometry among the three sides of a right triangle: $a^2 + b^2 = c^2$, where \(c\) is the hypotenuse.

Trigonometric Functions: Functions related to angles, commonly including sine, cosine, and tangent used in various formulas in trigonometry.