Problem 69 - Algebra and Trigonometry

Question

State the Law of Cosines in words.

Step-by-Step Solution

Answer

The Law of Cosines states: The square of one side of a triangle is equal to the sum of the squares of the other two sides minus two times the product of these sides and the cosine of the included angle.

Step 1: Identify the Law of Cosines

The Law of Cosines is a formula used to relate the lengths of the sides of a triangle to the cosine of one of its angles.

Step 2: Express the Law of Cosines in Words

The formula states that for any triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.

Step 3: Form the Complete Statement

In any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus two times the product of the other two sides and the cosine of the included angle.

Key Concepts

trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is essential for solving problems involving triangles, especially when the angles and sides are not perpendicular. Key concepts in trigonometry include sine, cosine, and tangent, which relate angles to side lengths in right-angled triangles. For non-right triangles, the Laws of Sines and Cosines are critical.

triangle sides

In a triangle, each side has a unique relationship with the other sides and angles. Knowing the lengths of two sides and the angle between them can help you find the length of the third side. Using the Law of Cosines, you can determine an unknown side or angle within any triangle, not just right-angled triangles. This law is very useful in various applications, including physics, engineering, and astronomy.

angle cosine relationship

The cosine of an angle in a triangle describes the relationship between the lengths of the sides of the triangle. According to the Law of Cosines, for any triangle with sides a, b, and c and an included angle C: equation: \[c^2 = a^2 + b^2 - 2ab \cos(C)\]This formula shows that the square of side c is the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. This relationship is extremely helpful for finding unknown side lengths or angles in a triangle.

geometric formulas

Geometric formulas are equations that relate various dimensions of geometric shapes. In the context of triangles, the most common formulas are the Law of Sines and the Law of Cosines. These formulas allow you to solve for unknown angles and sides. The Law of Cosines is particularly useful when dealing with non-right triangles. By understanding and applying these formulas, you can find missing side lengths and angles, providing a comprehensive understanding of triangle properties.