Law of Cosines

Definition

The law of cosines is a fundamental relationship in trigonometry that allows for the calculation of the third side of a triangle given the lengths of the other two sides and the angle between them. It is a generalization of the Pythagorean theorem, which only applies to right triangles.

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 any unknown side or angle in a non-right triangle, given the lengths of the other two sides and the included angle.
3. The law of cosines is particularly useful when solving trigonometric equations that involve non-right triangles, as it provides a way to relate the sides and angles.
4. The law of cosines is a generalization of the Pythagorean theorem, which only applies to right triangles, where $\cos(C) = 0$.
5. The law of cosines is one of the fundamental tools in trigonometry, along with the law of sines, for solving problems involving non-right triangles.

Review Questions

• Explain how the law of cosines is a generalization of the Pythagorean theorem and describe the relationship between the two.
  • The Pythagorean theorem is a special case of the law of cosines, where the triangle is a right triangle. In a right triangle, the angle between the two known sides is 90 degrees, so $\cos(C) = 0$. Substituting this into the law of cosines equation, we get $c^2 = a^2 + b^2$, which is the Pythagorean theorem. The law of cosines extends the Pythagorean theorem to non-right triangles by incorporating the angle between the known sides, allowing for the calculation of the third side.
• Explain how the law of cosines can be used to solve trigonometric equations involving non-right triangles.
  • The law of cosines provides a way to relate the sides and angles of a non-right triangle, which is essential for solving trigonometric equations in these types of triangles. By rearranging the law of cosines equation, we can isolate the unknown angle or side and use it to solve the equation. This allows us to determine the missing information in the triangle, which is crucial for solving trigonometric problems that involve non-right triangles, such as finding unknown angles or sides.
• Discuss the practical applications of the law of cosines in real-world situations.
  • The law of cosines has numerous practical applications in various fields, such as surveying, navigation, and engineering. In surveying, it can be used to determine the distance between two points when the lengths of two sides of a triangle and the angle between them are known. In navigation, the law of cosines is used to calculate the distance between two locations on the Earth's surface, given their latitudes and longitudes. In engineering, the law of cosines is employed in the design and analysis of structures, such as trusses and bridges, where the relationships between the sides and angles of non-right triangles are crucial for ensuring structural integrity.

Related terms

Trigonometry: The branch of mathematics that deals with the relationships between the sides and angles of triangles.

Pythagorean Theorem: A fundamental theorem in geometry that relates the lengths of the sides of a right triangle.

Non-right Triangles: Triangles that do not have a 90-degree angle, requiring the use of the law of cosines for calculations.