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. 3A/2

Explanation

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

\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Here, we need to rearrange this formula to solve for \( \cos(C) \):

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

### Step 1: Identify the values
We have:
- \( a = 8 \)
- \( b = 10 \)
- \( c = 12 \)

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

### Step 3: Substitute the values into the cosine formula
\[ \cos(C) = \frac{64 + 100 - 144}{2 \cdot 8 \cdot 10} \]

### Step 4: Simplify the numerator
\[ 64 + 100 - 144 = 164 - 144 = 20 \]

### Step 5: Calculate the denominator
\[ 2 \cdot 8 \cdot 10 = 160 \]

### Step 6: Substitute back into the cosine formula
\[ \cos(C) = \frac{20}{160} = \frac{1}{8} \]

### Step 7: Find angle \( C \)
To find angle \( C \), we take the inverse cosine:
\[ C = \cos^{-1}\left(\frac{1}{8}\right) \]

### Step 8: Calculate \( C \) using a calculator
Using a calculator, we find:
\[ C \approx 82.82^\circ \]

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

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