Question

what is the area of triangle def? round to the nearest tenth of a square unit. 10.3 square units 18.0 square units 20.0 square units 20.6 square units trigonometric area formula: area = $\frac{1}{2}absin(c)$

Explanation:

Step1: Identify the values of a, b and C

In triangle DEF, let $a = 5$, $b = 8$, and $C=180^{\circ}-(116^{\circ}+31^{\circ}) = 33^{\circ}$.

Step2: Apply the trigonometric - area formula

The trigonometric area formula is $A=\frac{1}{2}ab\sin(C)$. Substitute $a = 5$, $b = 8$, and $C = 33^{\circ}$ into the formula. We know that $\sin(33^{\circ})\approx0.5446$. Then $A=\frac{1}{2}\times5\times8\times\sin(33^{\circ})$.

Step3: Calculate the area

$A=\frac{1}{2}\times5\times8\times0.5446= 10.892\approx10.9$ square units. But if we assume the included - angle $C$ is $116^{\circ}$ (since the formula $\frac{1}{2}ab\sin(C)$ uses the included - angle between sides $a$ and $b$), then $A=\frac{1}{2}\times5\times8\times\sin(116^{\circ})$. Since $\sin(116^{\circ})\approx0.8988$, $A=\frac{1}{2}\times5\times8\times0.8988 = 17.976\approx18.0$ square units.

Answer:

18.0 square units