To find the length of side \( a \) in triangle \( ABC \) using the law of cosines, we can use the formula:
\[ a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \]
Given:
- \( b = 32 , \text{in} \)
- \( c = 45 , \text{in} \)
- \( A = 56^\circ \)
First, we calculate \( b^2 \) and \( c^2 \):
\[ b^2 = 32^2 = 1024, \] \[ c^2 = 45^2 = 2025. \]
Now we calculate \( 2bc \):
\[ 2bc = 2 \cdot 32 \cdot 45 = 2880. \]
Next, we calculate \( \cos(56^\circ) \):
Using a calculator,
\[ \cos(56^\circ) \approx 0.5592. \]
Now we can substitute the values into the law of cosines formula:
\[ a^2 = 1024 + 2025 - 2880 \cdot 0.5592. \]
Calculating \( 2880 \cdot 0.5592 \):
\[ 2880 \cdot 0.5592 \approx 1612.736. \]
Now substitute this back into the equation:
\[ a^2 = 1024 + 2025 - 1612.736. \]
Calculating \( 1024 + 2025 \):
\[ 1024 + 2025 = 3049. \]
Then:
\[ a^2 = 3049 - 1612.736 \approx 1436.264. \]
Now take the square root to find \( a \):
\[ a \approx \sqrt{1436.264} \approx 37.9 , \text{in}. \]
Thus, the length of side \( a \) is approximately:
\[ \boxed{37.9 , \text{in}} \]
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