In a `DeltaABC`, if a = 3, b = 5 and c = 7, find cos c

Options

A. `(1)/(2)`
B. `(1)/(sqrt(2))`
C. `-(1)/(2)`
D. 1

Answer

To find \( \cos C \) in triangle \( ABC \) with sides \( a = 3 \), \( b = 5 \), and \( c = 7 \), we can use the cosine rule. The cosine rule states that:

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step-by-Step Solution:

1. **Identify the values of the sides:**
- \( a = 3 \)
- \( b = 5 \)
- \( c = 7 \)

2. **Substitute the values into the cosine rule formula:**
\[ \cos C = \frac{3^2 + 5^2 - 7^2}{2 \cdot 3 \cdot 5} \]

3. **Calculate \( a^2 \), \( b^2 \), and \( c^2 \):**
- \( 3^2 = 9 \)
- \( 5^2 = 25 \)
- \( 7^2 = 49 \)

4. **Substitute these values back into the equation:**
\[ \cos C = \frac{9 + 25 - 49}{2 \cdot 3 \cdot 5} \]

5. **Simplify the numerator:**
- \( 9 + 25 = 34 \)
- \( 34 - 49 = -15 \)

So, the numerator becomes:
\[ -15 \]

6. **Calculate the denominator:**
\[ 2 \cdot 3 \cdot 5 = 30 \]

7. **Now substitute the values into the equation:**
\[ \cos C = \frac{-15}{30} \]

8. **Simplify the fraction:**
\[ \cos C = -\frac{1}{2} \]

### Final Answer:
Thus, \( \cos C = -\frac{1}{2} \).

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