In a triangle ABC, A = 8, b = 10 and c = 12. What is the angle C equal to ?

Options

A. `A//2`
B. 2A
C. 3A
D. none of these

Answer

To find the angle \( C \) in triangle \( ABC \) where \( A = 8 \), \( b = 10 \), and \( c = 12 \), we can use the Law of Cosines. The Law of Cosines states that:

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

Here, we need to identify the sides correctly. In this case, we have:
- \( a = 8 \) (side opposite angle \( A \))
- \( b = 10 \) (side opposite angle \( B \))
- \( c = 12 \) (side opposite angle \( C \))

Now, we can substitute the values into the formula:

1. **Calculate \( a^2 \), \( b^2 \), and \( c^2 \)**:
\[ a^2 = 8^2 = 64 \]
\[ b^2 = 10^2 = 100 \]
\[ c^2 = 12^2 = 144 \]

2. **Substitute into the Law of Cosines**:
\[ \cos C = \frac{64 + 100 - 144}{2 \cdot 10 \cdot 8} \]

3. **Simplify the numerator**:
\[ \cos C = \frac{164 - 144}{160} = \frac{20}{160} = \frac{1}{8} \]

4. **Find angle \( C \)**:
To find \( C \), we take the inverse cosine:
\[ C = \cos^{-1}\left(\frac{1}{8}\right) \]

5. **Calculate the angle**:
Using a calculator or trigonometric tables, we find:
\[ C \approx 82.82^\circ \]

Thus, the angle \( C \) is approximately \( 82.82^\circ \).