To find the area of triangle ABC using the trigonometric formula, we can use the 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 included angle.
Given:
- \( a = 7 \) inches,
- \( b = 12 \) inches,
- \( \angle C = 77^\circ \).
Plugging in the values:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times \sin(77^\circ) \]
Calculating the sine of \( 77^\circ \):
\[ \sin(77^\circ) \approx 0.974 \]
Now substituting this value back into the area formula:
\[ \text{Area} = \frac{1}{2} \times 7 \times 12 \times 0.974 \approx \frac{1}{2} \times 7 \times 12 \times 0.974 \]
Calculating it step by step:
\[ \frac{1}{2} \times 7 \times 12 = 42 \]
Now multiplying by \( 0.974 \):
\[ 42 \times 0.974 \approx 40.908 \]
Rounding to the nearest inch gives us:
\[ \text{Area} \approx 41 \text{ inches}^2 \]
Thus, the correct answer is:
41 inches²
1 answer