To find the area of triangle \( \triangle ABC \) using the trigonometric area formula, we can use the following formula:
\[ \text{Area} = \frac{1}{2}ab \sin(C) \]
where:
- \( a \) and \( b \) are the lengths of two sides of the triangle,
- \( C \) is the angle between those two sides.
Given:
- \( a = 7 \) inches,
- \( b = 12 \) inches,
- \( \angle C = 77^\circ \).
We first need to calculate \( \sin(77^\circ) \). Using a calculator, we find:
\[ \sin(77^\circ) \approx 0.9744 \]
Now, substituting the values into the area formula:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times \sin(77^\circ) \]
Calculating further:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.9744 \]
Calculating \( 7 \times 12 = 84 \):
\[ \text{Area} = \frac{1}{2} \times 84 \times 0.9744 \]
Now, \( \frac{1}{2} \times 84 = 42 \):
\[ \text{Area} = 42 \times 0.9744 \approx 40.973 \]
Rounding to the nearest inch:
\[ \text{Area} \approx 41 \text{ square inches} \]
Thus, the area of triangle \( \triangle ABC \) is approximately \( \boxed{41} \) square inches.