Question

If oblique △ABC has side b=32 inches, side c=45 inches, and included ∠A=56° , what is the length of side a ?(1 point)
Responses

38 in.
38 in.

2,440 in.
2,440 in.

1,439 in.
1,439 in.

49 in.
49 in.

Answer 1

To find the length of side \( a \) in triangle \( ABC \) using the Law of Cosines, you can use the following formula:

\[
a^2 = b^2 + c^2 - 2bc \cdot \cos(A)
\]

Where:
- \( a \) is the side opposite angle \( A \)
- \( b = 32 \) inches (side opposite angle \( B \))
- \( c = 45 \) inches (side opposite angle \( C \))
- \( A = 56^\circ \)

Let's calculate it step by step:

1. Calculate \( b^2 \):
\[
b^2 = 32^2 = 1024
\]

2. Calculate \( c^2 \):
\[
c^2 = 45^2 = 2025
\]

3. Calculate \( 2bc \):
\[
2bc = 2 \cdot 32 \cdot 45 = 2880
\]

4. Calculate \( \cos(56^\circ) \):
Using a calculator:
\[
\cos(56^\circ) \approx 0.5592
\]

5. Calculate \( 2bc \cdot \cos(A) \):
\[
2bc \cdot \cos(56^\circ) \approx 2880 \cdot 0.5592 \approx 1614.3
\]

6. Now substitute these values into the Law of Cosines formula:
\[
a^2 = 1024 + 2025 - 1614.3
\]
\[
a^2 \approx 1024 + 2025 - 1614.3 \approx 2434.7
\]

7. Now, take the square root to find \( a \):
\[
a \approx \sqrt{2434.7} \approx 49.34
\]

Therefore, the length of side \( a \) is approximately \( 49 \) inches.

The correct response is:
**49 in.**