Question

In a triangle ABC, with usual notations $a = 2$, $b = 3$, $c = 5$, then $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} =$

• A. $\frac{19}{30}$
• B. $\frac{19}{60}$
• C. $\frac{23}{60}$
• D. $\frac{38}{35}$

Hint:

Always simplify your algebraic expressions using variables before plugging in numbers! Working through the general formula first reduces three separate calculations down to a single clean expression, $\frac{a^2 + b^2 + c^2}{2abc}$, preventing calculation errors on your scratch pad.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the three side lengths $a$, $b$, and $c$ of a triangle $ABC$. We need to calculate the value of the algebraic sum containing cosine ratios divided by their respective side lengths.

Step 2: Key Formula or Approach:
According to the Law of Cosines in trigonometry, the angles can be expressed in terms of the sides as follows: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$$ Substitute these expressions into the target formula and simplify the fractions using a common denominator.

Step 3: Detailed Explanation:
1. Substitute each Law of Cosines identity into the given expression: $$\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{b^2 + c^2 - a^2}{2abc} + \frac{c^2 + a^2 - b^2}{2abc} + \frac{a^2 + b^2 - c^2}{2abc}$$ 2. Since all three components share the exact same denominator $2abc$, combine their numerators into a single fraction: $$= \frac{(b^2 + c^2 - a^2) + (c^2 + a^2 - b^2) + (a^2 + b^2 - c^2)}{2abc}$$ 3. Simplify the numerator by canceling out terms with opposite signs ($-a^2+a^2$, $-b^2+b^2$, $-c^2+c^2$): $$= \frac{a^2 + b^2 + c^2}{2abc}$$ 4. Substitute the given side magnitudes ($a = 2$, $b = 3$, $c = 5$) into this simplified expression: $$\text{Numerator} = 2^2 + 3^2 + 5^2 = 4 + 9 + 25 = 38$$ $$\text{Denominator} = 2 \times (2 \times 3 \times 5) = 2 \times 30 = 60$$ 5. Combine the terms to find the final reduced fraction: $$\text{Value} = \frac{38}{60} = \frac{19}{30}$$

Step 4: Final Answer:
The value of the expression is $\frac{19}{30}$, which corresponds to option (A).