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 answer

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.