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 (1 point)
options: 41 inches², 19 inches², 82 inches², 9 inches²

Explanation:

Step1: Recall the trigonometric area formula for a triangle

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

Step2: Substitute the given values into the formula

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

Step3: Calculate the area

Multiply \( 42 \) by \( 0.9744 \):
\[
A\approx42\times0.9744 = 40.9248\approx41
\]

Answer:

41 inches²