Question

If oblique ΔABC has side b=32 in., side c=45 in., and included ∠A=56° , then what is the length of side a?

Answer 1

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:

\[
\boxed{37.9 \, \text{in}}
\]