Question

Use the Law of Cosines to prove the identity $$ \frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c} $$

Short answer

By applying the Law of Cosines and simplifying, the identity is proved as \(\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}\).

Step-by-step solution

- Law of Cosines

Start with the Law of Cosines for each angle in the triangle. The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab\cos C \] \[ b^2 = a^2 + c^2 - 2ac\cos B \] \[ a^2 = b^2 + c^2 - 2bc\cos A \]

- Solve for Cosines

Solve each equation from Step 1 for \(\cos C\), \(\cos B\), and \(\cos A\) respectively:\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \] \[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \] \[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]

- Substitute Cosines

Substitute \(\cos A\), \(\cos B\), and \(\cos C\) from Step 2 into the left-hand side of the identity:\[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{1}{a} \left( \frac{b^2 + c^2 - a^2}{2bc} \right) + \frac{1}{b} \left( \frac{a^2 + c^2 - b^2}{2ac} \right) + \frac{1}{c} \left( \frac{a^2 + b^2 - c^2}{2ab} \right) \]

- Simplify

Simplify the expression from Step 3:\[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{b^2 + c^2 - a^2}{2abc} + \frac{a^2 + c^2 - b^2}{2abc} + \frac{a^2 + b^2 - c^2}{2abc} \]Combine all terms:\[ \frac{b^2 + c^2 - a^2 + a^2 + c^2 - b^2 + a^2 + b^2 - c^2}{2abc} = \frac{2(a^2 + b^2 + c^2)}{2abc} = \frac{a^2 + b^2 + c^2}{abc} \]

- Finalize

Multiply the numerator and denominator by \(\frac{1}{2}\):\[ \frac{a^2 + b^2 + c^2}{abc} \times \frac{1}{2} = \frac{a^2 + b^2 + c^2}{2abc} \]Thus, the identity is proved:\[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc} \]

Key concepts

Law of Cosines

The Law of Cosines is an essential rule in triangle geometry. It relates the lengths of the sides of a triangle to the cosine of one of its angles. This law is valuable especially for non-right triangles. The formula for the Law of Cosines is:\[ c^2 = a^2 + b^2 - 2ab\cos C \]It's important because it generalizes the Pythagorean theorem. For any triangle, if you know two sides and the included angle, you can find the third side.To explore further, consider the Law expressed for the other sides:\[ b^2 = a^2 + c^2 - 2ac\cos B \]\[ a^2 = b^2 + c^2 - 2bc\cos A \]By deriving these formulas, you can solve for the unknown side or angle, which can then be used in further trigonometric calculations.

Trigonometric Identities

Trigonometric identities are mathematical equalities that involve trigonometric functions and are true for every value of the occurring variables. In this problem, we use identities to simplify and prove the given formula involving cosines.One key identity within the context of the Law of Cosines is how we solve for the cosine of an angle:\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]Similarly for other angles:\[ \cos B = \frac{a^2 + c^2 - b^2}{2ac} \]\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc} \]These identities are used to transform complex trigonometric expressions into more manageable forms, allowing us to simplify the problem step by step.

Cosine Function

The cosine function is a fundamental trigonometric function. It is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. For any angle \(A\) in a right triangle:\[ \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \]However, in non-right triangles and for solving general problems, we use the Law of Cosines to find the cosine value of an angle. This function is key to understanding and solving various trigonometric equations, such as those found in the given exercise. The beauty of the cosine function is how it simplifies calculations involving angles and lengths in triangles.In the context of our problem, once we solve for \( \cos A \), \( \cos B \), and \( \cos C \), these values help substitute and simplify the overall expression.

Triangle Geometry

Understanding the basic properties and geometry of triangles is fundamental for solving problems using the Law of Cosines. Triangles have three sides and three angles, and the sum of the interior angles always equals 180 degrees.Here are some primary triangle characteristics to keep in mind:

• The sum of the lengths of any two sides must be greater than the length of the third side.
• Triangles can be classified by their side lengths (e.g., isosceles, equilateral, scalene) or their angles (e.g., acute, obtuse, right).

In our exercise, we deal with an arbitrary triangle, which means none of the sides or angles are assumed to be equal. By applying the Law of Cosines, we bridge the gap between the side lengths and angles, allowing us to solve the expression proving the identity.The goal is to use geometric relationships alongside algebraic manipulations to transform our given conditions into the required proof.