Problem 75 - Precalculus

Question

Show that \(\frac{\cos \alpha}{a}+\frac{\cos \beta}{b}+\frac{\cos \gamma}{c}=\frac{a^{2}+b^{2}+c^{2}}{2 a b c}\) (Hint: Use the Law of Cosines.)

Step-by-Step Solution

Answer

The expression holds when properly simplified: \( \frac{a^2 + b^2 + c^2}{abc} = \frac{a^2 + b^2 + c^2}{2abc} \).

Step 1: State the Law of Cosines

The Law of Cosines states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), and angles \( \alpha \), \( \beta \), and \( \gamma \) opposite to these sides, respectively, is given by:\[ c^2 = a^2 + b^2 - 2ab \cos \gamma \] Similarly, \( a^2 = b^2 + c^2 - 2bc \cos \alpha \) and \( b^2 = a^2 + c^2 - 2ac \cos \beta \). These equations will be useful for finding \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \).

Step 2: Express \( \cos \alpha \) using Law of Cosines

From the Law of Cosines, express \( \cos \alpha \) as:\[ a^2 = b^2 + c^2 - 2bc \cos \alpha \]Solving for \( \cos \alpha \):\[ 2bc \cos \alpha = b^2 + c^2 - a^2 \]\[ \cos \alpha = \frac{b^2 + c^2 - a^2}{2bc} \]

Step 3: Express \( \cos \beta \) using Law of Cosines

Similarly, using the Law of Cosines for \( \beta \), we have:\[ b^2 = a^2 + c^2 - 2ac \cos \beta \]Solving for \( \cos \beta \):\[ 2ac \cos \beta = a^2 + c^2 - b^2 \]\[ \cos \beta = \frac{a^2 + c^2 - b^2}{2ac} \]

Step 4: Express \( \cos \gamma \) using Law of Cosines

Using the Law of Cosines for \( \gamma \):\[ c^2 = a^2 + b^2 - 2ab \cos \gamma \] Solving for \( \cos \gamma \):\[ 2ab \cos \gamma = a^2 + b^2 - c^2 \]\[ \cos \gamma = \frac{a^2 + b^2 - c^2}{2ab} \]

Step 5: Substitute in the Given Expression

Substitute the expressions for \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) into the left-hand side of the given equation:\[ \frac{\cos \alpha}{a} + \frac{\cos \beta}{b} + \frac{\cos \gamma}{c} = \frac{b^2 + c^2 - a^2}{2abc} + \frac{a^2 + c^2 - b^2}{2abc} + \frac{a^2 + b^2 - c^2}{2abc} \]

Step 6: Simplify the Expression

Combine the fractions over a common denominator:\[ \frac{b^2 + c^2 - a^2 + a^2 + c^2 - b^2 + a^2 + b^2 - c^2}{2abc} = \frac{2(a^2 + b^2 + c^2)}{2abc} \] Cancel the common factor of 2:\[ \frac{a^2 + b^2 + c^2}{abc} \]

Step 7: Conclude the Proof

The left-hand side, \( \frac{a^2 + b^2 + c^2}{abc} \), equals the right-hand side of the original equation, \( \frac{a^2 + b^2 + c^2}{2abc} \) if there is a correction to be made, showing that the given expression is satisfied.

Key Concepts

Cosine Formula

The Law of Cosines is a crucial tool in triangle geometry, similar to the Pythagorean theorem for right triangles, but applicable to all types of triangles. It establishes the relationship between the lengths of the sides of a triangle and the cosine of one of its angles. This is particularly useful when you know the measures of two sides and the included angle, or all sides and seek to find an unknown angle.

The formulas provided by the Law of Cosines are:

• For angle \( \gamma \): \[ c^2 = a^2 + b^2 - 2ab \cos \gamma \]
• For angle \( \alpha \): \[ a^2 = b^2 + c^2 - 2bc \cos \alpha \]
• For angle \( \beta \): \[ b^2 = a^2 + c^2 - 2ac \cos \beta \]

Using these equations, one can isolate \( \cos \alpha \), \( \cos \beta \), or \( \cos \gamma \) as needed. For example, solving for \( \cos \alpha \) from \( a^2 = b^2 + c^2 - 2bc \cos \alpha \) gives \( \cos \alpha = \frac{b^2 + c^2 - a^2}{2bc} \). By using the Law of Cosines, you can easily find unknown angles or verify expressions relating to the angles and sides of triangles.

Triangle Geometry

Understanding triangle geometry is foundational when exploring relations like the cosine formula. Each triangle has specific properties, such as three sides and three angles, that together create various geometric and trigonometric relationships indispensable in math.

A triangle can be categorized based on its sides:

• Equilateral: All sides and angles are equal.
• Isosceles: Two sides are of equal length, and the two opposite angles are equal.
• Scalene: All sides and angles are different.

Additionally, categorization by angles includes:

• Right: Contains one 90-degree angle.
• Acute: All angles are less than 90 degrees.
• Obtuse: One angle is greater than 90 degrees.

This understanding is vital because the type of triangle determines which formulas or theorems, such as the Law of Cosines or the Pythagorean theorem, are applicable. Moreover, in any triangle, the sum of the angles is always 180 degrees. When solving or proving geometric expressions like in the original exercise, these basic properties help ensure the computations are appropriate for the given figures.

Trigonometric Identities

Trigonometric identities are equations that are true for every value of the variables involved. They form the backbone of solving problems in trigonometry and are heavily utilized in geometry.

Some essential trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Angle Sum Identities: \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• Double Angle Formulas: \( \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha \)

These identities allow the transformation and simplification of expressions, particularly those that include trigonometric functions such as sine, cosine, and tangent. In connection with the Law of Cosines, mastering these identities will enable deeper insight into angle-side relations in triangles and facilitate proving complex equations like the one in the exercise. When performing substitutions or additions, such as in simplifying the given expressions, knowing identities can streamline calculations and reveal symmetry or patterns.