Generalized Pythagorean Theorem

Definition

Generalized Pythagorean Theorem, also known as the Law of Cosines, is used to find a side or angle in any triangle given two sides and the included angle or three sides. It extends the Pythagorean Theorem to non-right triangles.

5 Must Know Facts For Your Next Test

1. The formula for the Generalized Pythagorean Theorem is $c^2 = a^2 + b^2 - 2ab \cos(C)$, where $a$, $b$, and $c$ are the lengths of the sides and $C$ is the included angle.
2. It can be rearranged to solve for angles: $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$.
3. This theorem is essential when dealing with obtuse or acute triangles.
4. In right triangles, it simplifies to the traditional Pythagorean Theorem because $\cos(90^\circ) = 0$.
5. It is pivotal in solving real-world problems including navigation, physics, and engineering.

Review Questions

• What is the formula for the Generalized Pythagorean Theorem?
• How does the Law of Cosines relate to the traditional Pythagorean Theorem in right triangles?
• When would you use the Generalized Pythagorean Theorem instead of the Law of Sines?

Related terms

Law of Sines: A trigonometric principle that relates the ratios of each side of a triangle to their respective opposite angles: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

Pythagorean Theorem: A mathematical principle stating that in a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Included Angle: The angle formed between two given sides of a triangle.