Law of Cosines

Definition

The law of cosines is a fundamental trigonometric identity that relates the lengths of the sides of a triangle to the cosine of one of its angles. It provides a way to solve for unknown sides or angles in non-right triangles, which are triangles that do not have a 90-degree angle.

5 Must Know Facts For Your Next Test

1. The law of cosines states that in a triangle with sides $a$, $b$, and $c$, and angle $C$ opposite side $c$, the following equation holds: $c^2 = a^2 + b^2 - 2ab\cos(C)$.
2. The law of cosines can be used to solve for unknown sides or angles in non-right triangles when at least one side and two angles, or two sides and one angle, are known.
3. The law of cosines is particularly useful when solving trigonometric equations that involve non-right triangles, as it provides a way to find unknown values.
4. The law of cosines is an extension of the Pythagorean theorem, which only applies to right triangles, and allows for the solution of more complex triangles.
5. Understanding the law of cosines is crucial for solving problems involving non-right triangles, which are commonly encountered in various fields, such as engineering, physics, and navigation.

Review Questions

• Explain how the law of cosines can be used to solve for unknown sides or angles in a non-right triangle.
  • The law of cosines states that in a triangle with sides $a$, $b$, and $c$, and angle $C$ opposite side $c$, the following equation holds: $c^2 = a^2 + b^2 - 2ab\cos(C)$. This equation can be rearranged to solve for any of the unknown sides or angles in a non-right triangle, provided that at least one side and two angles, or two sides and one angle, are known. By plugging in the known values and solving the equation, you can determine the missing side or angle in the triangle.
• Describe the relationship between the law of cosines and the Pythagorean theorem, and explain how the law of cosines extends the Pythagorean theorem to non-right triangles.
  • The law of cosines is an extension of the Pythagorean theorem, which only applies to right triangles. The Pythagorean theorem states that in a right triangle with sides $a$, $b$, and $c$ (where $c$ is the hypotenuse), the following equation holds: $c^2 = a^2 + b^2$. The law of cosines generalizes this relationship to any triangle, regardless of whether it is a right triangle or not. By including the cosine of the angle opposite the side $c$, the law of cosines allows for the solution of more complex, non-right triangles, where the Pythagorean theorem is not applicable.
• Analyze the importance of understanding the law of cosines in the context of solving trigonometric equations involving non-right triangles, and explain how it can be applied in various fields.
  • Understanding the law of cosines is crucial for solving trigonometric equations that involve non-right triangles, as it provides a way to find unknown sides or angles in these more complex geometric structures. The law of cosines is particularly useful in fields such as engineering, physics, and navigation, where non-right triangles are commonly encountered. For example, in engineering, the law of cosines can be used to calculate the forces acting on structures or the stresses in materials. In physics, it can be applied to problems involving the motion of objects in non-rectangular coordinate systems. In navigation, the law of cosines can be used to determine the position of a vessel or aircraft relative to known landmarks or coordinates. By mastering the law of cosines, you can expand your problem-solving capabilities and apply it to a wide range of real-world scenarios.

Related terms

Non-right Triangle: A triangle that does not have a 90-degree angle, where all three angles are acute or one angle is obtuse.

Trigonometric Equation: An equation that involves trigonometric functions, such as sine, cosine, or tangent, and is used to find unknown angles or sides of a triangle.

Cosine Function: A trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle, used to find the angle between two sides of a triangle.