Question

what is the area of triangle abc? round to the nearest tenth of a square unit.
trigonometric area formula: area = \\(\frac{1}{2}ab\sin(c)\\)
\\(\bigcirc\\) 54.9 square units
\\(\bigcirc\\) 58.8 square units
\\(\bigcirc\\) 61.8 square units
\\(\bigcirc\\) 64.1 square units

Explanation:

Step1: Identify sides and angle

We have side \( a = 13 \), side \( b = 10 \), and angle \( C = 72^\circ \) (since angle at \( C \) is \( 72^\circ \)). The trigonometric area formula is \( \text{Area} = \frac{1}{2}ab\sin(C) \).

Step2: Substitute values into formula

Substitute \( a = 13 \), \( b = 10 \), and \( \sin(72^\circ) \) into the formula:
\[
\text{Area} = \frac{1}{2} \times 13 \times 10 \times \sin(72^\circ)
\]

Step3: Calculate the value

First, calculate \( \frac{1}{2} \times 13 \times 10 = 65 \). Then, \( \sin(72^\circ) \approx 0.9511 \). Multiply these together: \( 65 \times 0.9511 \approx 61.8215 \). Rounding to the nearest tenth gives approximately \( 61.8 \).

Answer:

61.8 square units