To find the area of triangle \( ABC \) using the trigonometry area formula, we can use the following formula:
\[ \text{Area} = \frac{1}{2}ab \sin(C) \]
where:
- \( a = 7 \) inches
- \( b = 12 \) inches
- \( C = 77^\circ \)
Now, we can plug in the values:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times \sin(77^\circ) \]
First, let's calculate \( \sin(77^\circ) \):
\[ \sin(77^\circ) \approx 0.974 \]
Now, substituting this back into the area formula:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.974 \] \[ = \frac{1}{2} \times 84 \times 0.974 \] \[ = 42 \times 0.974 \approx 40.908 \]
Rounding this to the nearest inch, we find that the area is approximately \( 41 \) inches\(^2\).
So the answer is:
41 inches².
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