Key Equations
| Law of Cosines | [latex]\begin{array}{l}{a}^{2}={b}^{2}+{c}^{2}-2bc\cos \alpha \hfill \\ {b}^{2}={a}^{2}+{c}^{2}-2ac\cos \beta \hfill \\ {c}^{2}={a}^{2}+{b}^{2}-2abcos\gamma \hfill \end{array}[/latex] |
| Heron’s formula | [latex]\begin{array}{l}\text{ Area}=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\hfill \\ \text{where }s=\frac{\left(a+b+c\right)}{2}\hfill \end{array}[/latex] |
Key Concepts
- The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles.
- The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. Dropping an imaginary perpendicular splits the oblique triangle into two right triangles or forms one right triangle, which allows sides to be related and measurements to be calculated.
- The Law of Cosines is useful for many types of applied problems. The first step in solving such problems is generally to draw a sketch of the problem presented. If the information given fits one of the three models (the three equations), then apply the Law of Cosines to find a solution.
- Heron’s formula allows the calculation of area in oblique triangles. All three sides must be known to apply Heron’s formula.
Glossary
- Law of Cosines
- states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle
- Generalized Pythagorean Theorem
- an extension of the Law of Cosines; relates the sides of an oblique triangle and is used for SAS and SSS triangles