Question

use the image to answer the question. use the trigonometry area formula to find the area (to the nearest inch) of △abc, given a = 7 inches, b = 12 inches, and ∠c = 77 degrees (2 point) 41 inches² 18 inches² 82 inches² 9 inches²

Explanation:

Step1: Recall the trigonometric area formula

The formula for the area of a triangle using two sides and the included angle is \( A=\frac{1}{2}ab\sin C \), where \( a \) and \( b \) are two sides and \( C \) is the included angle between them.
Here, \( a = 7 \) inches, \( b = 12 \) inches, and \( C = 77^\circ \).

Step2: Substitute the values into the formula

First, calculate \( \sin(77^\circ) \). Using a calculator, \( \sin(77^\circ)\approx0.9744 \).
Then, substitute \( a = 7 \), \( b = 12 \), and \( \sin C\approx0.9744 \) into the formula:
\( A=\frac{1}{2}\times7\times12\times\sin(77^\circ) \)
\( A = 42\times0.9744 \)

Step3: Calculate the result

\( 42\times0.9744 = 40.9248 \approx 41 \) (rounded to the nearest inch).

Answer:

41 inches²